A Simple Intro duction

to Computable Analysis

Klaus Weihrauch

FernUniversitat

D Hagen

corrected nd version July

Contents

Intro duction

Computability on nite and innite words naming systems

Computability on the real numb ers

Eective representation of the real numb ers

Op en and compact sets

Continuous functions

Determination of zeros

Computation time and lo okahead on

Computational complexity of real functionss

Other approaches to eective analysis

App endix

References

Intro duction

During the last years an extensive theory of computability and computational

complexity has b een develop ed see eg Rogers Rog Odifreddi Odi Weih

rauchWei Hop croft and Ullmann HU Wagner and Wechsung WW

Without doubt this Typ e theory mo dels the b ehaviour of real world computers

for computations on discrete sets like natural numb ers nite words nite graphs

etc quite adequately

A large part of computers however is used for solving numerical problems Therefore

convincing theoretical foundations are indisp ensible also for computable analysis

Several theories for studying asp ects of eectivity in analysis havebeendevelop ed

in the past see chapter Although each of these approaches has its merits none

of them has b een accepted by the ma jority of mathematicians or computer scientists

Compared with Typ e computability foundations of computable analysis havebeen

neglected in research and almost disregarded in teaching

Thispaperisanintro duction to Typ e Theory of Eectivity TTE TTE is

one among the existing theories of eective analysis It extends ordinary Typ e

and connects it with abstract analysis Its origin is a denition

of computable real functions given by Grzegorczyk in Grz which is based on

the denition of computable op erators on the set of sequences of natural numb ers

Real numb ers are enco ded by fast with sp eed n converging Cauchy sequences

of rational numb ers and these are enco ded by sequences of natural numb ers A

real function is computable in Grzegorczyks sense i it can b e represented bya

computable op erator on such enco dings of real numb ers In the following years this

kind of computabilit y has b een investigated by several authors eg Grzegorczyk

Grz Klaua Kla Hauck Hau Hau and Wiedmer Wie The

computational complexity theory for real functions develop ed byKoandFriedmann

KF Ko can b e considered as a sp ecial branch

The study of representations ie functions from onto sets as ob jects of separate

interest results in an essential generalization of Grzegorczyks original denition and

admits to nd and justify natural computability denitions for functions on most of

the sets used in ordinary analysis Basic concepts are explained in Weihrauchand

Kreitz WK Wei KW Wei The theory has b een expanded in several

pap ers by Hertling Kreitz Muller and Weihrauch ranging from top ological conside

rations to investigation of concrete computational complexity WK KW Mue

Mue WK Wei Wei Wei A Wei B HW As an interesting

feature continuitycanbeinterpreted in this context as a very fundamental kind of

eectivity or constructivity and simple top ological considerations explain a number

of well known observations from eective analysis very satisfactorily

This pap er is not a complete presentation of TTE but only a technically and con

ceptually simplied selection from Wei The main stress is put on basic concepts

and on simple but typical applications while the theoretical background is reduced

to the bare essentials

We assume that the reader has some basic knowledge in computability theory Tu

ring machines computable functions recursive sets recursively enumerable sets

Intro duction

There are several go o d intro ductions eg the classical b o ok by Hop croft and Ull

man HU or Bridges Bri In addition to standard Calculus we use some simple

concepts from top ology top ological space op en and closed sets continuous func

tions metric space Cauchy sequence compact set Anyintro duction to top ology

eg Eng may b e used as a reference

In the following we axplain some notations which will b e used in this pap er By

f X Y wedenotea from X to Y ie a function from a

subset of X called the domain of f domf to Y The function f X Y is

totalidomf X in this case we write f X Y as usual A nite alphab et is

a nonempty nite set In Section denotes any nite alphab et with f g

In the following section is some xed suciently large nite alphab et containing

all the symb ols we shall need Let f g b e the set of natural numb ers

As usual is the set of all nite words a a with k and a a The

k k

emptyword is denoted by Let fa a j a g fp j p g b e the

 i

set of innite sequences or sequences with elements from We use suggestive

informal notations for dening nite and innite sequences over If u a a

k

v b b and p c c a b c then uv a a b b

l  i i i k l

m

up a a c c u a a a a a a m times u

k  k k k

uuu a a a a Ifx uv w and q uv p then u

k k

is a prex and v is a subword of x and q We extend the ab ove notations to sets of

fx j is a prex of xg and nite or innite sequences For example

u fp j u is a subword of pg

In Chapter we generalize computability from nite to innite sequences of sym

b ols and illustrate the denition bya numb er of examples Weintro duce the Cantor

top ology and show that computable functions are continuous Weintro duce nota

tions and representations and dene how top ological and computational concepts

are transferred from sequences to named sets In Chapter weintro duce standard

representations of the real numb ers the interval representation and the Cauchy re

presentation and investigate the computability concepts induced by them on the

real numbers Wegive examples for computable and noncomputable real numb ers

wecharacterize the recursively enumerable subsets of IR and prove computabilityof

anumb er of real functions In Chapter we give reasons for selecting the interval

and the Cauchy representation and for rejecting eg the decimal representation

W eprove that every computable real function is continuous weformulate the the

sis that every physically computable function is continuous and we provethatno

injective and no surjective representation can b e equivalent to the Cauchy represen

tation In Chapter weintro duce representations of the op en and of the compact

subsets of the real numb ers We prove eectiveversions of some well known classical

prop erties esp ecially weprove a computational version of the HeineBorel theorem

on compact sets Weintro duce representations of the classes C IRand C of

continuous real functions and discuss their eectivity in Chapter We presentsome

computational versions of well known prop erties and consider the determination of

a mo dulus of continuity of the maximum value the derivative and the integral

Determination of zeros of continuous functions is considered in Chapter Weprove

that the general problem can not even b e solved continuously Under certain restric

tions wehave a computable but nonextensional solution op erator A computable

op erator exists only on the set of continuous functions whichhave exactly one zero

In Chapter weintro duce as new concepts computation time and input lo okahead

of Typ e machines with innite output In Chapter wedenethemodiedbinary

representation which is appropriate for intro ducing computational complexity of real

functions We determine b ounds of time and input lo okahead for addition multipli

cation and by an application of Newtons metho d for inversion Finally we dene

the complexity of compact sets which can b e interpreted as plotter complexity

Some other approaches to eective analysis are discussed in Chapter

Computability on Finite and Innite Words Naming Systems

Computability on Finite and Innite

Words Naming Systems

In this Chapter is any nite alphab et ie any nite non emptysetTuring machi

nes are a convenient mathematical mo del for dening computabilityofwordfunctions

k

f By the ChurchTuring thesis a word function is computable

informally or bya physical device if and only if it can b e computed bya Turing

machine Moreover Turing machines mo del time and storage complexityofphysical

computers rather realistically A standard reference is the b o ok by Hop croft and

Ullman HU

In this section weintro duce our basic computational mo del for computable analysis

the Type machinesWeformulate a generalization of the ChurchTuring thesis

we prove that computable functions on nite or innite sequences are continuous

and dene recursively enumerable sets Weintro duce notations and representations

and dene how eectivity of elements sets functions and relations is transferred

by naming systems Many examples illustrate the denitions

Roughly sp eaking a Typ e machine is a for which not only nite

but also innite sequences of symb ols may b e considered as inputs or outputs We

give an informal denition of Typ e machines and their semantics

Denition Typ e machines

ATyp e machine M is dened bytwo comp onents

i a Turing machine with k oneway input tap es k a single oneway

output tap e and nitely manywork tap es

ii a typ e sp ecication Y Y Y with fY Y gf g

k   k

The typ e sp ecication expresses that f Y Y Y is the typ e of

M k 

the function computed by the machine M It tells which of the input and output

tap es are provided for nite and which for innite sequences Notice that input and

output tap es are restricted to oneway left to right

Denition semantics of Type machines

The function f Y Y Y computed by the Typ e machine

M k 

M the semantics of M is dened as follows

Case Y nite output



f y y w iM with input y y halts with

M k k

result w on the output tap e

Case Y innite output



f y y p iM with input y y computes

M k k

forever writing the sequence p on the output tap e

Notice that in the case Y the result f y y is undened if the machine

 M k

writes only nitely many symb ols on the output tap e but do es not halt A Typ e

machine can b e visualized by its underlying Turing machine

y

k input tap es

y

k

C

C

C

C

C

CW

work tap es

M

R

y

output tap e

f y y y

M k

Readers not familiar with Turing machines nd a more detailed denition in Ap

p endix A

We use the Typ e machines for dening computability of functions f Y

Y Y fY Y gf g The following denition generalizes the

k   k

common denition of computable wordfunctions since in the sp ecial case Y



Y Typ e machines are ordinary Turing machines

k

Denition Typecomputability

Let b e a nite alphab et Assume fY Y g f g k A

 k

Y Y Y is computableif f for some Typ e function f

k  M

machine M A sequence y is a computable elementof Y i the place



function f fg Y with f y is computable 

Computability on Finite and Innite Words Naming Systems

Since Turing machines and their halting computations are nite they havephysi

cal realizations of course only if size and time do not exceed certain b ounds By

denition Typ e machines may require innite input and output tap es and may

p erform innite computations which cannot b e realized actual ly since innite tap es

do not exist and innite computations cannot b e completed in reality Notice howe

ver that for a computation of a Typ e machine any nite p ortion of the output can

b e obtained already from a nite initial part of the p ossibly innite computation

and for this only nite initial parts of the input tap es are relevant This means that

the b ehaviour of a Typ e machine can b e approximated adequately byitsbehavi

our in the nite In this sense also Typ e machines and their computations can

b e realized physically Therefore anyTyp e computable function may b e called

intuitively computable or physically computable

Instead of Typ e machines any other common computability mo del eg FORTRAN

or PASCAL programs may b e used for denition and study of the computable

functions f Y Y Y provided inputs and outputs are oneway

k 

nite or innite les of symb ols Merely the denition of computational complexity

dep ends crucially on the computability mo del Below we shall use Typ e machines

for this purp ose

Typ e machines can b e considered as a certain kind of oracle Turing machines

Rogers Rog Hop croft and Ullman HU Sev eral other computable functions

of higher typ es have b een intro duced eg enumeration op erators

z

partial recursive operators F PF PF partial recursive functions F

and F partial recursive functionals F PF fg

where PF ff j f g see Rogers Rog xx

and computable functions F WeihrauchWei Each of these

denitions can b e derived from our Typ e computability and vice versa by using

appropriate natural enco dings Therefore it is very likely that every intuitively

computable function f Y Y Y is computable bya Typ e machine

k 

The ab ove considerations supp ort the following generalization of the ChurchTuring

thesis

Generalized ChurchTuring Thesis

A function f Y Y Y Y Y f g is computable

k   k

informally or bya physical device if and only if it can b e computed bya

Typ e machine

LikeChurchs Thesis also this more comprehensive thesis cannot b e proved In

the denition of Typ e machines wehave restricted input and output tap es to b e

onewayFor input tap es and for output tap es with nite output this restriction

is inessential b ecause a tw oway input tap e can b e simulated by a oneway input

tap e and a work tap e and for halting computations a twoway output tap e can

be simulated bya work tap e and a oneway output tap e The oneway output for

innite computations however is an essential restriction see Example b elow

Among other prop osed basic computational mo dels for dening computabilityon

Typ e ob jects like etc the Typ e machines are particularly simple and

concrete they admit to explain the top ological connection b etween classical analysis

and computational theory in a very transparentway and moreover they admit a

direct denition of very realistic computational complexities as we shall show later

on We illustrate the denition of Typ e computabilityby several examples

Example

Let f g and dene f by

f div

i i

f p for all i and p

The following owchart copies the leftmost zeros from the input tap e to the output

tap e It halts i the input is not



    

R R







HALT

The owchart together with the typ e sp ecication denes a Typ e machine

which computes the function f

Example

Let f g and dene f by

div if fi j pig is nite else

f pn

if hp niseven

if hp nisodd

where hp n is the p osition of the n th one in p ie hp n is that number i for

i and car d fkij pk g n The following owchart together which p

with the typ e sp ecication denes a Typ e machine which computes the

function f

Computability on Finite and Innite Words Naming Systems



    

R R

 









    

R R





From nowonwe shall no longer sp ecify Turing machine owcharts in full detail but

give only informal descriptions Typ e sp ecications will b e given implicitly by the

context

Example

Consider the problem of dividing real numb ers by where the real numb ers are re

presented by innite decimal fractions decimal expansions The wellknown pap er

and p encil metho d by reading the input left to right and writing the output left to

right can b e programmed easily bya Typ e machine without work tap es The nth

output symbol b f g and the nth remainder r f g are determined

n n

by the symbol a f g and the previous remainder r f g as follows

n n

r a b r

n n n n

The sign and the decimal p ointmust only b e copied from the input to the output

tap e A owchart consisting of sequences one for each previous remainder

f g of consecutive tests one for each symbol f g plus write

statements etc solves the problem We omit a detailed owchart

Example

Consider the problem of multiplying real numb ers by where the real numb ers are

represented by innite decimal fractions The scho ol metho d for multiplying nite

decimal fractions adds intermediate results from right to left It is also p ossible to

p erform the addition from left to right In this case however from time to time

carries may app ear which run from right to left switching nines to zeros

right to left addition left to right addition

                   

 

 

   

 

 

 

 

        

 

   

This metho d with left to right addition can b e applied also to innite decimal

fractions It can b e implemented easily on a mo died Typ e machine whichhasa

twoway output tap e

hine multiplies innite decimal fractions by NowweshowthatnoTyp e mac

Assume that there is a Typ e machine M whichmuliplies innite decimal fractions

by Consider the input p ThenM must pro duce the output

q or the output q Consider the case q

There is a computation step in which M writes the rst symb ol on the output

tap e Up to this step M has read only the rst k symb ols for some k from the

k

input tap e Consider the input sequence q Since the rst k symbols of

q and q coincide also with input q the machine M will write the symb ol as the

k

rst output But since M is a multiplier it must write on the output tap e

This is a contradiction The case q is handled accordingly

Therefore no Typ e machine multiplies innite decimal fractions by Also the

more general problem of multiplying two real numb ers in decimal representation

cannot b e solved bya Typ e machine By Example twoway output is strictly more

powerful than oneway output We continue with examples for noncomputable

functions

Example

Let f g and dene f by

if p

f p

otherwise

We show that f is not computable Assume that some Typ e machine computes f

Consider the input p Then for some number k M will pro duce

k

Since the rst k symbols of the output in k steps Consider the input p

Computability on Finite and Innite Words Naming Systems

p and p coincide and since M can read in k steps at most k symb ols M halts with

output also for input p Since f p M cannot compute the function f

Example

The function f from Example has no computable prop er extension Assume on the

contrary that for some w the function f dened by f w

i i

f p for all i and p is computed by some Typ e machine M Then

with input for some k M will pro duce the output w in k steps Esp ecially

k

this implies lg w k Consider the input q Since the rst k symbols of

and q coincide and since M can read in k steps at most k symb ols M halts with

k

j

output w also for input q Since lg w k we obtain f q w f q

M

Therefore M cannot compute f

In the same way it can b e shown that also the function f from

Example has no computable prop er extension

In the pro ofs in Examples and only the following fundamental niteness

prop erty of computable functions f Y Y f g has b een used

Finiteness prop erty for computable functions

If f z y thenany nite prex of the output y is already determined by

some nite p ortion of the input z

This niteness prop erty is equivalentto continuity wrt the Cantor topology on

and the discrete topology on

Denition Cantor topology on discrete topology on

fA j A g is called the discrete topology on

d

fA j A g is called the Cantor topology on

C

is called the Cantor space over

C

A is op en ie A AsetU is op en ie U i Every set

d d C C

there is some A with p U w A w is a prex of p for all p

If p U already a nite prex w of p suces to prove this prop erty The top ology

dene the distance can b e generated from a metric spaceFor p q

d

if p q

n

dp q

where n is the length of the longest

common prex otherwise

It is easy to showthat d is a metric space A subset X is an op en ball

n

i it is a closed ball i X w for some word w w B w

n

Bcw where n lg w The set of op en balls fw j w g is a basis of

the Cantor top ology On cartesian pro ducts Y Y Y Y f g

C k k

we consider the pro duct top ologies

For functions f the niteness prop erty can b e formulated as follows

Assume f z y Then for any op en ball B y there is some op en ball B z

such that f B z B y But this means that f is continuous in z iethe

nitenness prop ertyisequivalent to continuity

Theorem computable continuous

Every computable function f Y Y Y Y is continuous

 k 

Pro of

Consider the case Y It suces to show that for Let f y y y

  k

any neighb ourho o d w of y there is some neighbourhood X of y y with

  k

f X w Let M be a Typemachine which computes f Letw be a

 

neighb ourho o d of y Then M with input y y writes the prex w of y

 k  

in nitely many steps During this computation only the prex w of the input y

i i

on Tap e i can b e read i k Then X w Y w Y is an op en

k k

neighb ourho o d of y y with f X w The case Y can b e proved

k  

similarly

Therefore for functions on and continuity is a necessary condition for compu

tability only continuous functions can b e computable Continuity ie the niteness

prop erty of functions is a very elementary constructivity prop erty In each of the ex

amples and wehaveproved discontinuity of the function under consideration

Of course there are also continuous functions which are not computable

Example

Let d b e a total function with r ang e d f g which is not computable

Then the functions f g and h are

continuous but not computable where

f w n dn for all w n

g div

k

g q dk for all k q

hq n dn for all q n

Computability on Finite and Innite Words Naming Systems

From a Typ e machine for f g or h one could construct a Turing machine computing

the function d

An imp ortant ob ject in ordinary recursion theory Rogers Rog is an eective

 

Godel numb ering P of the set P of the computable functions

f The theory of Typ e computability can b e deep ened byintro ducing

ab ab ab

eective notations P of the sets P of the computable functions

a b ab ab

f and eective representations F of certain sets

ab a b

F of continuous functions f a b fg Denitions and some

prop erties are given in App endix B For details see Wei These naming systems

however will not b e used in the following

The comp osition of computable functions is computable or has a computable exten

sion For simplicitywe consider only unary functions

Theorem composition of computable functions

Let f Y Y and g Y Y Y Y Y f g b e computa

    

ble

j

If Y Y then gf is computable

 

If Y Y then gf has a computable extension h suchthat

 

f dom h dom f dom gf dom

Pro of

Let M and M be Typ e machines computing f and g resp ectively It is p ossible

f g

to construct from M and M aTyp e machine M whichsimulates alternately

f g

the computations of M and M taking in turn the output symbols of M as the

f g f

input symb ols for M M is simulated until it requires the rst input symb ol M is

g g f

simulated until it pro duces the rst output symb ol M is simulated until it requires

g

the next input symb ol etc The computable function f has the desired prop erties

M

As a simple consequence of Theorem computable functions map computable

elements to computable elements A subset A is recursively enumerable re

i A dom f for some computable function f and A is recursive

or decidable i A and n A are re We generalize these basic denitions from

recursion theory as follows

Denition re and recursive sets

Consider k and Y Y f g

k

A set X Y Y is called recursively enumerable rei

k

X dom f for some computable function f Y Y

k

For U W Y Y we call U re in W iU W X for

k

some re set X

For U W Y Y wecallU recursive or decidable in W

k

i U and W n U are re in W

Assume M dom f where f Y Y for some Typ e machine

M M k

M This machine M is an abstract pro of system for the set X domf If

M

y X then M applied to input y halts The nite computation can b e considered

as a pro of for the prop ertyy X in this pro of system If y X then there is no

such a pro of

If y dom f then only a nite p ortion of the p ossible innite input y can b e

M

read by M during its nite computation Therefore any re set is op en Anyopen

set X has the form A for some A It is easy to showthatX

e subset A U is is re i X A for some re even for some recursiv

recursivein W i there is a computable function f Y Y with

k

W dom f and U f fgW ief y y U for all y W

The sets U recursivein are particularily simple U is recursivein i

U A for some nite set A This follows from compactness of

Finite or innite sequences of symb ols can b e used as names of other ob jects like

natural numb ers rational numb ers nite graphs rational matrices real numb ers

subsets of etc Examples are the binary notation of the natu

bin

ral numb ers and the decimal representation IR of the real numbers

dec

where is a suciently large alphab et Weintro duce naming systems and redu

cibilities for comparing them

Denition notations representations reducibility

A naming system of a set M is a notation or a representation of M

where a notation is a surjective function M naming by

nite strings and a representation is a surjective function

M naming by innite sequences

Y M with Y Y For functions Y M and

f g wecall reducible to iy dom y

f y for some computable function f Y Y We call and

equivalent i and

Topological reducibility and topological equivalence are dened

t t

accordingly by substituting continuous for computable

Computability on Finite and Innite Words Naming Systems

If f wemaysay that the function f translates the naming system into

the naming system examples translation from PASCAL to ASSEMBLER from

English to German For a naming system Y M there are informations

ab out the elements of M which can b e obtained computationally from their names

Translation cannot increase this information If and wemaysay that

names contain more computationally available information than names As an

example consider the following two notations and of Let A re and



not recursive Dene w w for all w w w if w A w w if

 

w A x div otherwise Then obviously but The rst symbol

  

of any name of a word w is the answer to the undecidable question w A



This is not the case for names We illustrate Denition byanexample

Example

Let d d and let b e an alphab et with f d g For any

a a d dene a notation of the natural numbers and a

a

representation R of the nonnegative real numb ers as follows where

a 

f a g

a

dom nfg

a

a

k 

a a a a a a a

a k  k  i a

dom

a

a a

k 

a a a a a a a a a a a

a k   k  i a

Let P fe j e is a prime factor of ag Then for any a b f dg the

a

following prop erties hold

a b

a b

ifP P

a b b a

if P P

a t b b a

There is a Turing machine which translates aadic numbers into badic numb ers ie

By symmetry wehave also hence The computable function

b b a a b a

f with f w w translates into Since bywe

b b a b

have It is not very dicult to design a Typ e machine which translates

a b

to if P P Consider e P n P Thene cb b

a b b a b a b

c forsome c bute has a unique name p for which neither

b b a

p nor p a As in Example or it can b e shown that there is no

continuous translator which is correct for input p Details are left to the reader

A naming system Y M transforms eectivity concepts from Y to M First

we dene computable p oints and op en re and recursive subsets

Denition

Let Y M i k b e naming systems

i i i

x M is computable i there is a computable element y Y with

y x

X M M is open rerecursive i

k k

fy y Y Y j y y X g

k k k k

is op en re recursiveindom dom

k

For any naming system Y M the set fX M j X is

op en g is called the nal topology of

Example

Let b e the binary notation of Every n is computable

bin bin

every subset A is op en A subset A is re i it is re Let be

bin bin dec

the representation of real numb ers by innite decimal fractions

Every rational number is computable For a pro of notice that the decimal

dec

names of the rational numb ers are p erio dic

p

computable A simple trial and error searchby squaring nite decimal is

dec

p

with p fractions yields a sequence p

dec

For any X IR X is op en X is op en

dec

Wesketch a pro of Let X IR b e op en Let p X Since X is op en

dec

n n

there is some n with p p X Let w

dec dec p

b e the prex of p containing the rst n digits after the decimal p oint Then

w dom X ie p w dom X This means

dec p dec p dec

dec

X is op en in dom Let X Therefore p has an op en neighb ourho o d in

dec dec

X be op en Consider x X and x wlg There is some p with

j

px such that q whenever p wq with w and q f g

dec

and q f X is op en in dom there are some w Since g

dec

dec

j

dom with p wq and w we obtain X Since q

dec

dec

x wq w w w X

dec dec dec dec

Computability on Finite and Innite Words Naming Systems

j

There is another p with p x and q whenever p wq with

dec

w and q f g As in the rst case we conclude that there is some

prex w of p with

x w w X

dec dec

Therefore x I X for some op en interval I This shows that X is op en

Notice that the nal top ology of Y M is indeed a top ology on M Next

we dene relativ e computability and continuity of functions and relations

Denition relatively eective relations and functions

For i k let Y M b e naming systems

i i i

A relation Q M M M is computable

k  k 

continuous i there is some computable continuous function f

Y Y Y with

k 

y y f y y Q

k k  k

whenever x y y x Q

k k

A function F M M M is computable

k  k 

continuous i there is some computable continuous function f

Y Y Y with

k 

F y y f y y

k k  k

whenever F y y exists

k k

Relative computability continuity of a relation Q can b e considered as an eec

tiveversion of the mere existence statementx x x Qx x x If

k  k 

Q is computable some computable function f transforms anyname

k 

y y ofx x from the domain of Q intoanamey of some x such

k k  

x Roughly sp eaking for eachx x we can determine that Qx x

k  k

some x with Qx x x If in y y y y

 k  k k k

k

implies f y f y y then there is a computable or continuous

 k 

k

function G M M M with x x Gx x Q Sucha

k  k l

function G is called a choice function of Q A relation Q may b e computable without

having a continuous choice function see Example b elow Denition is

a sp ecial case of where Q is singlevalued A function F is

k 

continuous i some transforms any name y y ofsome

k

x x domF into some name y of F x x Notice that by deni

k  k

tion every restriction F of F is continuous computable if F is

k 

continuous computable A Typ e machine computing some re

k 

lation Q or some function F actually transforms merely sequences of symbols it is

the user who interprets these sequences as names of ob jects

Example

Dene the enumeration representation En of the set of subsets of

by

n

n Enp is a subword of p

for all n and p and let b e the binary notation of We

bin

assume f g Then the following prop erties hold

A is Encomputable A is re

fA n j n Ag is En re

bin

j n Ag is not En op en fA n

bin

fA n j n Ag is En computable

bin

There is no En continuous function

bin

j

f with f A A for all A

Wesketch the pro ofs

This is a simple recursion theoretic exercise

Let M beaTyp e machine which for input p w searches in p for the subword

n

where n w M halts i suchasubword has b een found

bin

WehaveEn Q fA n j n Ag Assume Q is

bin  

k

En op en Then for some k Enq Q for all q But

bin bin 

k k

q and Enq Q

bin 

Let M beaTyp e machine with f which searches in input

M

m

p for the rst app earence of a subword If sucha word is found

w m is written on the output tap e then a word w with

bin

Supp ose there is some continuous function g with g p

bin

j

fEnp whenever En p Theng g By conti

k k

nuityof g there is some k with g fg g fgLet

k k

p and q Then g pfEnp

bin

fEnq g q contradiction f f g

bin

Notice that the set Q fA n j n Ag is En computable but has not even

bin

aEn continuous choice function f

bin

The computability concepts induced on sets by naming systems Def

remain unchanged if the naming systems are replaced by equivalent ones and cor

resp ondingly the induced top ological prop erties remain unchanged if the naming

systems are replaced by top ologically equivalent ones For the pro of only the fact

that the computable and the continuous functions are closed under comp osition is

needed

On the other hand nonequivalent naming systens induce dierent computability

Computability on Finite and Innite Words Naming Systems

theories on M Therefore the induced eectivity concepts on a set according to

Defs and dep end crucially on the underlying naming system

In TTE computability on a set M is intro duced in two steps

denition of computable functions on nite or innite sequences of symbols

denition of a naming system Y M

As for numb er functions we are not interested in arbitrary computability concepts

on M but only in those which meet some intuition which are natural In Step

which is indep endentof M wecho ose the Typ e computable functions whichare

eective by our generalized ChurchTuring thesis see Chapter Eectiveness

of a naming system of a set M can b e dened only relative to some structure on M

It is an essential feature of TTE that eectiveness of the intro duced naming systems

is justied by general principles

Computability on the Real Numb ers

In this chapter weintro duce standard naming systems for the natural the rational

and the real numb ers and study the induced computabilityWegive examples of

computable real numbers characterize the op en re and recursive sets and

C

prove computability of functions like addition and multiplication and of real analytic

functions with computable p ower series

From now on let b e a suciently large nite alphab et containing all the symbols we

shall need Let b e the onetoone binary notation without leading

bin

zeros of the natural numb ers and let b e the onetoone binary

Q

notation of the rational numb ers by signed reduced fractions of binary numb ers for

an exact denition see App endix C These notations and all the equivalent ones are

usually called eective Are there other eective notations of the natural und

rational numb ers In App endix C we show that the equivalence classes of and

bin

can b e characterized by simple eectivity prop erties and a maximality principle

Q

In the following text we shall use the abbreviations

u u for all u dom

bin bin

u u for all u dom

Q Q



As an example addition on is computable wrt ie f with

bin

f x y x y is computable in more detail there is a computable

bin bin bin

u v g u v forallu v Dom Also function g with f

bin bin

multiplication exp onentiation arithmetical subtraction and division minimum and

maximum are computable wrt On the rational numb ers addition subtrac

bin

tion multiplication division maximum and minimum are computable wrt

Q

The most p opular representation of the set IR of the real numb ers is that by in

nite decimal fractions decimal expansions IR Unfortunatelyvery

dec

simple functions like x x are not computable see Example This

dec dec

already indicates that in not adequate for a foundation of computabilityon IR

dec

since real number multiplication should b e computable Toovercome this problem

weintro duce two standard representations of IR the interval representation and the

Cauchy representation

Denition interval and Cauchy representations

Dene two representations IR interval representation and

I

IR Cauchy representation as follows

C

px i there are u v u v dom with

I   Q

p u v u v and x supu infv

  i i

i

i

px i there are u u dom with

C  Q

k

and x lim u u j u p u u k i kj

i k i 

i

Computability on the Real Numbers

If p u v u and p x then x is the only p oint in the intersection of all

  I

closed intervals u v If px with p u u then u isaCauchy

i i C  i i

i

sequence of rational numb ers converging to x with sp eed and consequently

k

j u xj for all k Therefore we can asso ciate with u u the sp ecial

k 

n n

u u Notice that we sequence I of nested intervals where I

n n n n n

consider only these fast converging Cauchy sequences as names First we compare

the three representations and

dec I C

Lemma relation between decimal and Cauchy representation

dec I C

C t dec

Pro of

A Typ e machine M can be constructed which with input

dec I

p a a a a dom f ga f g writes

k   dec i

i

u r and v r if u v u v on the output tap e where

i i  

i

u r and v r if and r Q is the rational value of

i i

the nite decimal fraction a a a a Then f translates into ie

k  i M dec I

p f p for all p dom

dec I M dec

Let M be a Typ e machine which with input p u v u v dom

I C   I

w on the output tap e where for i u v dom writes q w

i i Q 

the word w is determined as follows M searches for a pair of natural numb ers k m

i

i

with jv u j and then sets w v Sincep dom the searchmust b e

m k i m I

i i

successful its result guarantees w p w hence jw w j for all

i I i i n

ni Therefore p q

I C

LetM beaTyp e machine which with input p w w dom

C I  C

i i

u w and v w writes u v u v on the output tap e where

i i i i  

Then obviously p f p

C I M

Assume there is some continuous function f with p

C t dec C

f p for all p dom Since IR f p or

dec C C

f pConsiderthe case f p Since f is continuous there is some

n n n

n such that f Letu and q u Then

q but f q hence f q q Therefore f do es not

C dec C

translate q correctly The case f p is handled accordingly

By Lemma the decimalnames contain more continuously accessible information

than names Belowwe shall give convincing arguments that not the decimal

C

representation but the Cauchyrepresentation is adequate for dening computability

on the real line Since wemayusealso instead of for investigating

C I I C

computabilityon IR whenever appropriate

Convention

In the following and will b e our standard naming systems of QI and IR

bin Q C

resp ectivelyFor simplicity in connection with computable re and recursive

we shall omit prexes like etc and shall say computable instead

bin C Q

of computable re instead of re etc

bin bin Q C bin

By Denition the computable real numbers are by the ab oveconvention those

numb ers whichhave computable names or computable names Lemma

C I

Example computable real numbers

Every rational numb er is computable

Consider r QI Dene u by u r dene q uu u

Then q is computable and q r

C

p

is computable

 n 

Dene f by f n that k with k k Then f

n

u f n Thenp u u u is computable and is computable Let

n  

p

p

C

log is computable



k n k

Dene f by f n that k with Thenf is

computable Let u kn and v k nThenp u v u v is

n n  

computable and p log

I



i

For A dene x f j i AgThen

A

x is computable A is recursive

A

i

u f j i Assume that A is recursive For k dene u by

k k

A i k g Then p u u is computable with x p

 A C

Assume that x is computable For all w a a k a a f g

A  k  k

i

let x fa j i k gIfx x for some w then x is computable

w i A w A

j

By assumption x pforsome by Assume x x for all w

A C A w

computable p u u For any w f g there is some i with



i i

u x or x u In the rst case wehave x x inthe

i w w i A w

second case x x Therefore W fw j x x g is decidable Compute

w A w A

a sequence y y of words inductively by y if x x otherwise

  A

y y ifx x y otherwise Then is the last symbol of y i

k k y A k k

k

k A Therefore A is recursive

Computability on the Real Numbers

Further examples of computable real numb ers can b e obtained by applying compu

table functions to computable arguments see b elow The limit of any computable

sequence of computable real numb ers with computable modulus of convergence is

computable

Theorem limit of computable sequence with computable convergence

Let y bea computable sequence of real numb ers suchthat

i i bin C

n

i j mnjy y j for some computable function m

i j

m is called a computable modulus of convergence Then x lim y is

i

i

computable

Pro of

By assumption for any i j aword u dom can b e computed such that

ij Q

y u u Let v u for all i Then q v v is

i C i i i 

m i i 

computable For all kiwehave

j v v j ju y j jy y j jy u j

i k

m i i  m i  m i  m k  m k  m k k 

i i k k 

max

i

and

v xj ju y j jy xj j

i m i i  m i  m i 

i i

i

We obtain x q Therefore x is computable

C C

Example

i

Let A b e re but not recursive Dene x IR by x f j i AgBy

A A

Example x is not computable Since A is re and not recursive there is some

A

total injective computable function f with A r ang e f Obviously

f n f k 

x f j n g Dene a sequence s y by y f j k ng

A n n n

Then s is computable even computable and increasing Since

bin C bin Q

its limit x is not computable it cannot have a computable mo dulus of convergence

A

by Theorem The idea is from E Sp ecker Sp e

The set of computable real numb ers is a denumerable subset of IR however it cannot

be enumerated eectively We prove a p ositiveversion of this statement For every

computable enumeration of computable real numb ers a computable numb er which

is not enumerated can b e determined

Theorem

Let x be a computable sequence Then a

i i bin C

j

x with x x for all i can b e determined

i

Pro of

j

By diagonalization we construct a computable number x suchthatx x for all

i

i For any i we can determine a sequence q u u with x q

i i i i C i

i

therefore there is a computable function g with g u We

ii 

i i

obtain j g x j for all i We compute u v dom for

Q i i i Q



i as follows

i

u g v u

 Q  

Assume u and v have b een determined Dene

i i



i i i

u u v u if g u

i i i i Q i



i

u u v v otherwise

i i i i

i

v u x u v and u v u v The construction guarentees

i i i i i i i i i

j

Therefore x u v u v exists and x x for all i Additionallythe

I   i

sequence u v u v is computable hence x is computable ie

  I C

computable

Next wecharacterize the op en the re and the recursive subsets of IR

C

Theorem

For any X IR

X is op en X is op en

C

dom Y re X is re Y dom

Q C Q

S

X f u v j u v Y g

X is recursive X or X IR C

Computability on the Real Numbers

Pro of

Let X be op en Consider x X There are words u dom with

C i Q

i i i i

u u xu u i Weobtain

i i i i

X isopenindom there is some k with u u x Since

C C 

C

k k

u u X Sincex u u u u x

C  k k k C  k

has an op en neighbourhood in X Therefore X is op en On the other hand let X

b e op en Consider p u u X Since X is op en there is some i



C

i i

suchthat p p X For any q V u u u

C C  i

i i

dom weha ve ju q j therefore j q pj hence

C i C C C

q X Therefore V is an op en neighb ourho o d of p in X This shows

C

C

that X is op en

C

Let X be re Then there is some re set W with X W

C

C

dom Let M be a Typ e machine which for input u v works as

C

follows M searches systematically for some w W and words u u u

 k

dom such that cf of this pro of

Q

k k

u u u u u u

 k k 

w is a prex of u u u

 k

k k

u u v u

k k

M halts as so on as suchwords have b een found By the pro of of ab ove

S

Y domf has the desired prop erties On the other hand let X fu v j

M

u v Y g with re Y Let M beaTyp e machine which for input p

u u dom works as follows M searches systematically for some k

 C

k k

u v M halts i the searchhas and some u v Y with u u

k k

b een successful Obviously domf dom fp j p X g Therefore

M C C

X is re

C

If X is recursive X is re and IR n X is re Therefore X and IR n X

C C C

are op en and op en by and IR are the only sets X with this prop erty

C

since the real line is connected

By Theorem the nal top ology of is the usual top ology on IR generated

C IR

by the op en intervals By IR has no nontrivial recursive subsets No non

C

y of real numb ers can b e decided if only names are available The trivial prop ert

C

n

characterizations hold accordingly for X IR n



Lemma some re subsets of IR IR

Let a IR b e computable The sets

j

fx IR j x ag fx IR j xag fx IR j x ag

 

j

fx y IR j x yg fx y IR j x y g

are recursively enumerable

Pro of

We consider only the most general case x y Let M be a Typ e machine which

for input p q u u v v dom dom searches for some

  C C

k k

k with u v andhaltsassoonassucha k has b een found Then

k k

f p q exists i p q The other pro ofs are left to the reader

M C C

The complements of the ab ove sets fx j x ag etc are not re since they are

n

not op en Theorem The re subsets of IR are closed under nite union

and intersection By Lemma op en intervals with computable b oundaries are

re Let A b e re and not recursive Then the in terval x where x

A A

i

f j i Ag is re for a pro of use Theorem but by Example its upp er

b oundary is not computable Many of the functions studied in classical Analysis

are computable

Theorem some computable real functions

The real functions x y x y x y x y x y maxx y

and x x are computable

Let a be a computable sequence and let R be

i i bin C 

i

the radius of convergence of the p ower series a x For each R with

i

i

RR the real function f dened by f x a x if jxjR

 R R i

div otherwise is computable

Pro of

We use the fact that the given functions are continuous and that their restric

tions to QI which is dense in IR are computable

Q Q

x y

Let M beaTyp e machine which for input p q p q dom p

C

u u q v v writes the sequence r y y on the output

  

tap e such that

y u v

n n

n

for all n Let x p and y q For all nk wehave

C C

k k

jj jy y u u j jv v j

n k n k

n k

k k

j x y j j y u xj jv y j

n n

n

We obtain r f p q dom and r x y

M C C

Computability on the Real Numbers

x y

Let M be a Typ e machine which for input p q p q dom p

C

u u q v v writes the sequence r y y on the output

  

tap e such that

y u v

m n m n

n

for all n where m is the smallest natural numb er with

m m

u j and jv j j

 

Let x p and y q For all n wehave

C C

m

j u jju u j ju j

n n  

m

v j For all knwehave and corresp ondingly j

n

j y y j ju v u v j

m n m n m k m k

n k

u v v j jv u u j j

m n m n m k m k m n m k

m mn n

n

and corresp ondingly j y x y j We obtain r f p q dom and

M C

n

r x y

C

maxx y

Let M be a Typ e machine which for input p q p q dom p

C

u u q v v writes the sequence r y y on the output

  

tap e such that

y maxu v

n n

n

for all n Let x pandy q Assume knand s

C C

u v u v If s u then max

n n k k n

k

j y y j u maxu v u u

n k k n k

n k

k

v u v g we obtain jy y j By symmetry for the other cases s f

n k k

n k

k

y maxx y j is proved Therefore in the same way Corresp ondingly j

n

r f p q dom and r maxx y

M C

x

Let M be a Typ e machine which for input p u u dom and

 C

j

px works as follows First M searches for the rst N with

C

N

j u j Assoonassuchanumber N has b een found M writes v v

N 

N

v u for all k Since ju j for all on its output tap e where

k N k i

i N v exists for all k For all k n with kn we obtain

k

j v v j ju u j

k n N k N n

u u jju jju j j

N n N k N n N k

N k N N k

k

and corresp ondingly jv xj Therefore r f p dom and

k M C

r x

C

It suces to prove the theorem for rational numbers R Let R QI be some

rational number with RR R ByCauchys estimate there is some number



M with

i

ja j M R

i

for all i Given some x with jxjR for each n we shall approximate

N

P P

i i

a x b f x c b whereN is suciently large and b c c y d

i R i  N n

i i

are rational numb ers where jb xj and the ja c j are suciently small such

i i

n

that jf x d j LetM be a Typ e machine whichforany input

n

p u u dom with jxjR where x p generates a sequence

 C C

q v v v dom where v is computed as follows

 i Q n

M determines some N such that



N n

M RR R R R

M determines some b QI jbjR with



N

X

n

jx bj MR i

i

For any i N the machine M determines some c QI with

 i

i n

c jjbj N ja

i i

N

P

i

Dene v c b

n i

i

For all x b IR with jxj jbj R and all i wehave

i i i i i i

jx b j jx bjjx x b b jjx bji R

We obtain for jxjR

v j jf x

n R

N N N

P P P

i i i

jf x a x j j a x c b j

R i i i

i i i

N

P P

i i i

j a x j j a x c b j

i i i

i

iN

N N

P P P

i

i i i i i

ja x a b j ja b c b j M R R

i i i i

i i iN

N N

P P

i

i N

ja c jjbj M RR R R R ja jjx bj i R

i i i

i i

N

P

n n

jx bj i MR

i

n 

n

Computability on the Real Numbers

n

Consequently for all knwehave jv v j Therefore f x

k n R

v v

C 

i

In general for computable a the function f xa x is not computable on

i i i

fx jjxj R g where R is the radius of convergence

 

Example

There is some computable injective function h such that r ang eh

h n n

and A r ang eh is re but not recursive Dene c and a c

n n

n

n

for all n Then the sequence a is computable and the p ower series a x

n n n

n

has radius of convergence By Theorem f xa x is computable on

n

every interval r with r We show that f is not computable on

If f is computable on there is a computable function M with

k n

a M k Dene g by

n

g k maxfn j hn k g

We obtain for all k

k  g k 

M k a

g k 

hg k  k  g k 

k k  g k 

k g k 

therefore

k

g k log M k log

Since M log and division are computable see b elow g k H k for some com

putable function H We obtain k A n H k hn k by the

denition of g Therefore A must b e recursivecontradiction We conclude that

n

a x is not computable on

n

x

Since the p ower series for e sin x arctan x ln x etc are computable these

functions are computable by Theorem Since the computable real functions

are closed under comp osition many other real functions are computable at least on

appropriate subsets of their domainseg x xx y minx y x jxjany

p

y

x x ln xx y x p olynomial function with computable co ecients x

 

x y x y etc Notice that every restriction of a computable function

is computable Since computable real functions map computable real numb ers to



computable real numbers numb ers like e e arc sin ln e



p



etc are computable cos

The join of two computable real functions at a computable p oint is computable

Lemma join of two functions

Let f f IR IR b e computable functions let a IR b e computable



with f a f a Then f IR IR dened by



f x if x a

f x

f x otherwise



is computable

Pro of

We only sketch a pro of Consider i f g Since f is computable there is a Typ e

i

wrt the Cauchy representation For any input machine M which computes f

i C i

i

p and any n we can compute an interval I with rational b oundaries suchthat

pn

i

f p I

i C

pn

i

lim length I

pm

m

if p domf

C

There is some computable sequence q t t t dom with a q

 i Q C

Let M beaTyp e machine which for input p u u dom pro duces a

 C

sequence v w v w wherethewords v w dom are dened as follows

  n n Q

n n

I if u t

n n

pn

 n n

v w

I if t u

n n

n n

pn

J otherwise

pn



where J is the smallest interval containing I and I For all p domf we

pn C

pn pn

obtain f p f p therefore f is computable

C I M

Let us call a function f IR IR a polygon i there are real numbers

x y with x y

n n  

x x x

 n

Computability on the Real Numbers

and

div if xx or x x

 n

f x

y where x y is on the straight line connecting

x y andx y if x x x

i i i i i i

The p oints x y x y are called vertices of f As a corollary of lemma

  n n

we obtain that every p olygon function with computable vertices is computable



Computabilityonthe complex plane CI is dened byidentifying CI with IR For



there are two functions f f IR IR dened by any function f CI CI



f x iy f x y if x y The function f is called computable i f and f are

 

computable Computability of complex addition multiplication division z jz j

and z arg z follows from Theorem The pro of of Theorem can

easily b e generalized to complex p ower series Therefore also complex functions like

z z

sinz e w z w lnz etc are computable on appropriate subsets of their

domains

We conclude with an example of a computable binary relation which has no com

putable choice function

Example

Let S fx n IR jjx nj g Then S as a relation is computable more

precisely computable But S has no continuous choice function ie there

C bin

is no continuous function f IR with x f x S for all x IR

C bin

We prove b oth statements

Let M beaTypemachine which for input p u u dom determines

 C

u j Then j p f pj forallp some word w with j w

 C bin M bin

dom Therefore S is computable Notice that we cannot guarantee

C C bin

extensionalityof f iewe cannot guarantee f p f p if p

M bin M bin M C

p

C

Assume that there is some continuous function f IR with jx

C bin

f xj for all x IR Then there is some continuous function g

with j p g pj for all p dom Wehave f and f Let

C bin C

y inf fx j f xg There is some p u u dom suchthat py

 C C

and u u is a neighbourhood of y for all k Consider the case

C  k

g pBycontinuityof g there is some k with g u u f y Then

 k

fg There is some q u u with f q For this q wemust have

 k C

g q contradiction The case f y is treated accordingly

Eective Representation of the Real

Numbers

In Section wehaveintro duced ad ho c the Cauchy representation IR

C

of the real numb ers and studied the induced computabilityon IR Since we are not

interested in some arbitrary computability theory on IR we need a go o d justication

for the choice of the Cauchy representation or some equivalentone

In this section we explain why the Cauchy representation is top ologically natural

for the real line and why it is computationally natural We mention the concept

of admissible representations and formulate the imp ortantcontinuity theorem for

admissible representations Finally we explain why several other representations of

IR cannot b e natural

We assume without further discussion see App endix C that our notation

Q

QI of the rational numb ers induces the natural computabilityonQI Let K

t

b e the set of all representations of I R such that

fu v pj up v g is op en in dom

A representation is in K i

t

up v already a nite p ortion of p guarantees up v

or more formally

up v w w is a prex of p and u q v for all q w

This means that nite p ortions of p admit to lo cate p arbitrarily precisely by

rational numb ers from b elow and ab ove on the real line Representations not having

this prop erty dont seem to b e very useful In fact the Cauchy representation

C

the interval representation Def and also the decimal representation

I dec

are elements of K Since the class K do es not consist of a single t

t C t dec t

equivalence class but is distinguished by maximalityin K

C t

Theorem is eective for the real line

C

Let K b e the set of all functions IR suchthat

t

j u p v g is op en in dom fu v p

Then for any function IR

K is continuous

t t C

Thus is except for equivalence the unique p o orest continuous representation of C

Eective Representation of the Real Numbers

IR If px then all true prop erties of the form u x v and only these can

C

b e obtained from nite p ortions of any name p of x There is a surprising formal

C

similarity of Theorem to a well known theorem in recursion theory Wei Let

 

P b e some eectiveGodel numb ering of the set P of the partial



recursive functions f LetK b e the set of all numb erungs P



such that U fi x y j x y g is re Then K Notice

i

that U is re i satises the universal Turing machine theorem There is a

computational version of Theorem expressing that the Cauchy representation is

not only top ologically but also computationally sound

Theorem is computational ly eective

C

Let K b e the set of all functions IR such that

c

fu v p j u p v g is re in dom

Then for any function IR

K

c C

Thus is also maximal in the sub class K K wrt computable reducibility

C c t

Notice that wehave a denition of is continuous but no denition of is

computable As in Theorem corresp onds to the smntheorem and

to the utmtheorem

Pro of

Assume K By assumption there is a Typ e machine M which for anyinput

c 

up v Let M beaTyp e machine u v p dom halts i

which with input p tries to pro duce a sequence u u u dom as

 i Q

follows For computing u by an exhaustive search M tries to nd some u v m

n

n

with v u suchthatM with input u v p halts in at most



m steps If this search is successful M cho oses u uThen p f pfor

n C M

all p dom

Assume By assumption there is some Typ e machine M with p

C

f p for all p dom Let M be a Typ e machine which with input u v p

C M

u v dom p dom works as follows By simulating M with input pM

Q

generates the sequence f p u u and halts as so on as some n is found with

M 

n n

and uu u v Then

n n

fu v p j u p v g dom domf M

therefore K

c

Let b e the set of op en subsets of IR By Theorem the representation

IR

IR is admissible with nal top ology App endix D contains a short

C IR

denition of admissible representations Here we merely formulate the imp ortant

continuity theorem For a broad discussion see KW Wei Wei

Theorem continuity

For i k let M b e an admissible representation For

i i

any function F M M M wehave

k 

F is continuous F is continuous

k 

Since every computable function on is continuous and since is admissible by

C

Theorem every computable real function is continuous Because of its imp ort

ance we prove this fact directly without using Theorem

Theorem

Every computable real function is continuous

Pro of

Let f IR IR b e computable Then f is computable There is a Typ e

I C

machine M such that f p f p for all p domf Let O IR be open

I C M I

tervall I with and let f x O Wehavetoshow that f I O for some op en in

x I There are words u v dom with u u and v v such

i i Q  

that x p where p u v u v There are words w w dom

I    Q

such that f pq where q w w dom Since q f x O and

M  C C

O is op en there is some m such that w w O For pro ducing

C  m

w w the machine M reads at most u v u v for some k from the

 m   k k

u v Then there is some q suchthat x p where input tap e Let x

k k I

u v u v q By the b ehaviour of M f p w w hence p

  k k M  m

f p O We obtain f x f p f p O Therefore f I O and

C M I C M

u v x I for I

k k

Eective Representation of the Real Numbers

n

The general case f IR IR is proved accordingly

At rst glance very simple discontinuous real functions like the jump j x

if x otherwise or the Gauss bracket g x bxc integer part of x

are intuitively computable Clearly these two functions are easily denable in our

mathematical language but easily denable do es not mean computable This

solves the seeming contradiction

Some functions can b e made computable bycho osing appropriate representations

Consider a representation of the real numb ers such that p determines the sign

of x if p x Then of course the jump is computable

bin

Every function f IR IR can b e made computable for some appropriate

C

representation dep ending on f Dene pq pq x p

C

x and q f x This dirty trick cannot b e applied to twoplace functions

C

Lemma

There is no representation IR such that the test l IR IR

where l x y if x y otherwise is continuous

bin

Pro of

Assume that there is some continuous function f such that

l p q f p q Consider z p Wehave l pp f p p

bin bin

Therefore f p p Since f is continuous f w w fg for some

prex w of pFor any x y w we obtain x y hence fz g w Therefore

with fz g w This however is imp ossible for any z IR there is some w

car dIR since car d

Wemayinterpret the result as follows the function l is absolutely not computable

byphysical devices

According to Theorem the Cauchy representation is distinguished from other

representations of the real numb ers except for top ological equivalence where the

top ology on IR by the op en intervals with rational b oundaries is considered as

IR

the reference structure on IR Theorems and conrm that the Cauchy

representation induces the natural computability theory on the real line Since

the decimal representation is not even tequivalentto it is unnatural

dec C

Rememb er also that by Example the computable real function x x is

not computable Are there representations in the equivalence class of

dec C

dec

which are simpler than The next theorem excludes some obvious simplications

C

Theorem restrictions for admissible representations of IR

No total representation IR is tequivalentto

C

No injective representation IR is tequivalentto

C

Dene the naive Cauchy representation of IR by px i there

n

are u u dom withp u u and x limu

 Q  i

i

Then is not tequivalentto

n C

Pro of

One can showthat with the Cantor top ology is a compact metric space If

then is continuous Theorem Since anycontinuous function maps

t C

compact sets to compact sets also IR must b e compact but IR is not b ounded

Assume that there is an injective representation IR with

t C

By Theorem X IR is op en X is op en X is op en

C

We conclude that is a continuous function Any continuous function maps

is connected sets to connected sets The set IR is connected But IR dom

j

not connected Let p q IR p q Then there is some w with p w

and q w Let A w IR B n w IR Then A and B are

b oth op en in IR and nonemptyand IR A B and A B Hence

dom is not connected

Assume that there is some continuous function f with p f pfor

n C

all p dom Let p Then p Let f pu u By

n n 

k

continuityof f there is some k with f u u Then f is incorrect



k k

for p since f but p

C n

Op en and Compact Subsets

Op en and Compact Subsets

IR

Since the cardinalityof thepower set of IR is greater than the cardinalityof

it has no representation Therefore in our approachwe are not able to investigate

IR

computability of functions like f IR with f X y y supX We

restrict our attention to the op en subsets and to the compact subsets of IR which

have representations We dene a standard representation of showthatitis

IR

top ologically and computationally eective and list some computability results We

intro duce several eective representations of the set K IR of the compact subsets

of IR prove a computational version of the HeineBorel theorem and give examples

for computable op erations on the set K IR of compact sets

Denition representation of

IR

Dene a representation of the set of op en subsets of IR as follows

op IR

pX i there are words u v u v dom

op   Q

with u v for all i and p u v u v suchthat

i i  

S

u v j i g X f

i i

for all p and X

IR

We use the convention a a A sequence p is a name of X i

op

p enumerates a set of op en intervals with rational b oundaries which exhausts X

Since every op en subset X of IR is the union of a set of op en intervals with rational

b oundaries the ab ove function is surjective ie it is a representation of

op IR

The equivalence class of can b e dened by a simple eectivity prop ertyanda

op

maximality principle cf Theorems

Theorem eectivity of

op

Let K b e the set of all functions suchthat

t I R

fu v p j u v and u v pg is op en in dom

Then for all functions

IR

K

t t op

Let K b e the set of all functions such that

c IR

fu v p j u v and u v pg is re in dom

Then for all functions

IR

K

c op

Thus is except for equivalence the unique p o orest representations of for

op IR

whichevery true prop erty of the form u v X can b e obtained from a nite

p ortion of any name of X We omit a pro of of Theorem A few examples for

induced eectivity are listed in the following theorem

Theorem properties of

op

X is computable X is re

op C

fx X IR j x X g is re

IR C op

Union and intersection on are computable

IR op op op

For f IR IR dene H by H X f X forall

f IR IR f

X Then

IR

H is continuous if f is continuous

f op op

H is computable if f is computable

f op op

Pro of

This is an immediate consequence of Theorem

Let M be a Typ e machine which for inputs p w w dom and

 C

q u v u v dom works as follows M searches systematically for

  op

i i

indices i k with u w and w v M halts i such indices have

k i i k

b een found We obtain fp q j p q g domf dom dom

C op M C op

see Def

Consider only inputs of the form p u v u v dom and q

  op

w x w x dom For the case of union let M beaTyp e machine

  op

u v w x For the which pro duces from p and q the output u v w x

   

case of intersection let M beaTypemachine which pro duces a list of all

u v for which there are numb ers i k with u v u v w x intervals

i i k k

This follows from the more general Theorem b elow

A subset X IR is compactiX is closed and b ounded By the HeineBorel

theorem X is compact i for every set of op en subsets of IR with X

IR

there is some nite subset with X Thecharacterization remains valid

if ab ove is replaced by the set of all op en intervals with rational b oundaries The

IR

following four representations can b e derived from these characterizations

Op en and Compact Subsets

Denition representations of the compact sets

Let K IR b e the set of all compact subsets of IR Dene a notation of the

nite sets of op en intervals with rational b oundaries by

w i there are words u v u v dom with

k k Q

w u v u v and f u v u v g

k k k k

Dene representations and of K IR asfollows

c cb w

closed representation

pX i pIR n X

c op

closed b ounded representation

pX i there are u dom and q dom with

cb bin op

p uq X IR n q and X u u

op

weak covering representation

pX i there are words w w dom with

w 

j j

p w c w c such that for all w dom



S

X w iw w

i

strong covering representation

j j

pX i p w c w c asabovesuch that



S

j

w and I w I X g fw w g fw j X



If pX then p enumerates the complementof X If pX then p gives a

c cb

b ound of X and enumerates the complementof X If pX then p enumerates

w

all coverings of X with nitely manyopenintervals with rational b oundaries In the

case of instead of only the minimal coverings are enumerated by names

w

The reducibilities b etween the four ab ove representations are given by the following

theorem

Theorem computational HeineBorel theorem

cb c c t cb

computational HeineBorel theorem

w cb

w w t

Pro of

There is a Typ e machine M with f uu u u u for

M  

u u u dom Then f translates to If pX thenno

 Q M cb c c

nite prex w of p contains any information ab out a b ound of X hence

c t cb

More formally assume that there is some continuous function f

with p f p for all p dom There is some p u v u v

c cb c  

with pIR n pfg The sequence f p has the form uu v By

c op

 

continuityof f there is some prex w of p with f w u But there is

some q w dom with q u u contradiction

c c

We show Assume pX Thenp enumerates all coverings of

w cb w

X with nitely manyintervals with rational b oundaries From the rst such

covering a b ound for X can b e determined easilyFrom the other coverings

one can determine an enumeration of op en intervals with rational b oundaries

which exhausts IR n X Weprove this more formallyLet be some



standard bijectivenumb ering of There is a Typ e machine which for input

j j

p w c w c dom pro duces a sequence uu v u v dom

 w   cb

with the following prop erties

w u u



k

u v if i uv with



u v

u v w

i i

k

otherwise

Then p f p for all p dom

w cb M w

We show Assume pX Then from p weknow some closed

cb w cb

interval I with X I and an enumeration I I of op en intervals exhausting



IR n X Since I is compact

X w iI w I I for some k

 k

Therefore we can enumerate all words w with X w Weprovethismore

formally There is a Typ e machine which for input p uu v u v

 

j j

dom pro duces a sequence q w c w c dom with the following

cb  w

prop erties

k

j

w if i c w with



w

u v w u v u v

i

  k k

w otherwise

where u u w Then p f p for all p dom

cb w M cb

Since is a restriction of is trivial follows from Theorem

w w w t

b elow

The equivalence can b e considered as a computational version of the

w cb

Op en and Compact Subsets

HeineBorel theorem There is an improvement of for which a

cb

stronger computational version of the HeineBorel theorem can b e proved see KW

Every compact subset of IR has a maximum and a minimum the compact sets are

closed under union and intersection and f X is compact if f is continuous and X

is compact Eectiveversions of these facts are listed in the following theorem

Theorem computable operations on compact sets

The function max K IR IR is computable but not

C

continuous

w C

Intersection and union are computable and

w w w

computable

For f IR I R dene H K IR K IR by H X f X for

f f

all X K IR Then

H is continuous and continuous if f is continuous

f w w

H is computable and computable if f is compu

f w w

table

Pro of

j j

There is a Typ e machine M which transforms any p w c w c dom



v dom where the u v are dened as follows into q u v u

I i i  

u v is the greatest interval in w

i i i

wrt the order u v u v vv or v v and u u Then

max p f p Assume there is a continuous function f

I M

j j

with max p f p for all p dom There is some p x c x c with

w C w 

Then f p and u Since f is p Let f pu u

C  w 

j j j j

continuous there is some i with f x c x c c x c u u u There

 i  

j j j j

is some q x c x c c x c with q fgWeobtainmax q but

 i w w

f q contradiction

C

The pro of is left to the reader

This follows from the more general Theorem b elow

The denitions and theorems of this section can b e generalized easily from IR to

n

the ndimensional Euklidean space IR There are theorems similar to Theorem

characterizing the eectivity of and of We dont go into more details

w

here We mention without pro of that is admissible and that the nal top ology

fX K IR j X is op en in domg of the representation is the Hausdor

top ology on the set K IR of the compact subsets of IR Eng Wei Esp ecially

Theorem is applicable to

Representations of Continuous Real Functions

Representations of Continuous Real

Functions

Let us denote by C X the set ff IR IR j f continuous and domf X gWe

intro duce explicitly standard represenations of C IR and of C and give

IR C

sucient reasons for their eectivity As examples we consider mo dulus of continuity

maximum dierentiation and integration

Denition representation of C IR

Dene a representation C IR as follows

IR

pf i there are words u v x y dom i

IR i i i i Q

j j

with p u v x y c u v x y c

   

such that for all rational numb ers a b c d

u b v c x y y f a b c d ia

i i i

i

for all p and f C IR



Roughly sp eaking pf i p enumerates all a b c d QI with f a b c d

IR

This representation has the following remarkable eectivity prop erty cf Thms

Theorem is eective

IR

Let L b e the set of all functions C IR such that the func

t

tion appl y C IR IR IR where appl y f x f x is

C C

continuous Then

L

t t IR

for all functions C IR

Let L b e the set of all functions C IR such that appl y is

c

computable Then

C C

L

c IR

for all functions C IR

Again there is a formal similarity with the characterization of eectiveGodel num



b erings of P

satises the univ ersal Turing machine theorem

see the remarks after Theorem We omit a pro of of Theorem Esp ecially

wehave L ie the universal function appl y C IR IR IR of

IR c IR

is computable Some interesting prop erties are listed in the following

IR C C

theorem

Theorem some computable operations

f IR IR is computable f is computable

C C IR

The function H C IR dened by H f X f X is

IR IR

computable

IR op op

The function G C IR K IR K IR dened by Gf X f X

is computable and computable

IR w w IR

The comp osition F C IR C IR C IR dened by F f g

f g is computable

IR IR IR

wn that continuous functions are uni We do not prove this theorem It is wellkno

formly continuous on compact subsets We shall prove a computable version of this

theorem We call a function m a modulus of continuity of a function

f IR IR on X domf i for all x y X and n

m n n

jx y j jf x f y j

For the set fm j m g we use the following standard representation

pm p u u withi u mi

 bin i

for all p and m

Theorem determination of a modulus of continuity

There is a computable function h such that hp z

is a mo dulus of continuityof pon z z for all p dom and all

IR IR

z dom

bin

Pro of

Consider N If f IR IR is continuous for any x N N andany n

there are numbers a b cd QI suchthatx a b and f a b c dand

x x x x x x

n

d c ObviouslyN N f a b j x N N g Since N N

x x

is compact a nite subset of intervals suces for covering N N Therefore for

n there is a nite set of quadrup els a b c d of rational numb ers i

i i i i

k suchthatN N a b a b and f a b c d and

k k i i i i

n

for i k Let c minfb a j i kg Assume d c

i i i i

Representations of Continuous Real Functions

N x y N and jx y j c Then there are i j with x a b y

i i

j

a b and a b a b Consequently f x c d f y c d and

j j i i j j i i j j

n

j

c d c d Therefore jf x f y j LetM beaTyp e machine

i i j j

j j

which for input p t c t c dom where t u v x y and z dom

 IR i i i i i bin

pro duces a sequence q w w where w is dened as follows M searches

 n

n

for a nite set I of indices such that y x for all i I and

i

i

z z fu v j i I g Such a set I exists M determines m with

i i

m

v u j i I g Then w dom is determined by w m minf

i i n bin bin n

By the ab ove considerations q is a mo dulus of continuityof f pon z z

IR

Denition and Theorems and can b e easily generalized from C IR to

n

C X where X IR is re Also generalizations from IR to IR are straightforward

Next we study the class C of the continuous functions f IR IR with

domf Weintro duce a metric on C and dene as a generalization of

a standard Cauchy representation of C

C

For f g C dene the distance df gmaxfjf x g xjj x g

C d is a metric space Let Pg b e the set of all p olygon functions f C

with rational vertices It is known that Pg is dense in C d ie for any f

n

C and n thereissomeg Pg with df g The op en ball B f a

with centre f C and radius a can b e visualized by a strip e of width a

surrounding f

Denition Cauchy representation of C

Dene a notation Pg of the set Pg of all p olygon functions

with rational vertices from C by

w g i there are u v u v dom with

  k k Q

v w u v u

k   k

u u and g is the p olygon

 k

with the vertices u v u v

  k k

Dene a representation C of C by

C

pf i there are w w domwith

C 

k

j j

p w c w c k i kdw w

 i k

k

and k df w

k

Similar to the denition of the Cauchy representation of the real numbers we

C

consider in only fast converging Cauchy sequences of rational p olygon functions

k

we can also write f lim w If as names Instead of k df w

k k

k

j j

w c w c f then the graph of f is the intersection of all the closed balls

C 

k

Bcw The representation is equivalent to the representation obtained

k

from C IRby restricting the domains from IR to

IR

Theorem

Dene C by

px if x

IR

px

div otherwise

for all p and x IR

Then

C

As a consequence Theorem holds accordingly for instead of C has

C IR

other imp ortant dense subsets eg the p olynomial functions with rational co e

cients or the trigonometric p olynomials with rational co ecients Standard notati

ons of these dense subsets induce Cauchy representations which are equivalentto

C

For the functions from C a mo dulus of continuity can b e computed from their

names We mention without pro of that the representation is admissible where

C C

the nal top ology is generated by the op en balls of the metric space C d As

a consequence the continuity theorem Theorem can b e applied to

C

Corollary modulus of continuity

There is a computable function g such that g pisa

mo dulus of continuityof p on for all p dom

C C

Pro of

By Theorem there is a computable function f with p f p

C

The mo dulus of continuityof f p on is a mo dulus of continuityof p

IR C

on Dene g phf p with h from Theorem

For a function f C the number y maxff x j x g is called the

maximum value of f andany x with f xy is called a maximum point of f

For functions f from C the maximum values can b e determined eectively

Determination of maximum p oints will b e reduced to the determination of zeros in Chapter

Representations of Continuous Real Functions

Theorem determination of maximum

The function Max C IR dened by Maxf maxff x j

x g is computable

C C

Pro of

j j

Let M be a Typ e machine which for input p w c w c dom determines

 C

a sequence u u where u Maxw Let f p f xMaxf

 n n C

f w f x Maxf Then for any n

n n n n n

n n n

f x f x f x f x f x

n n n n n n

n

u Maxf j We obtain Maxf u u therefore j

n C 

Esp ecially the maximum value of a computable function f C is computable

We close this section with some remarks on dierentiation and integration By the

next theorem dierentiation on the set C of the continuously dierentiable

functions from C cannot b e p erformed eectivelyif is used as the naming

C

system

Theorem noneectivity of dierentiation

The dierentiation op erator Dif f C C dened by

Dif f f g i g is the derivativeof f for all f g C is not

continuous

C C

Pro of

Assume that Dif f is continuous Since the continuity theorem can b e

C C

applied to Dif f must b e continuous But this is false Consider the functions

C

f f f C dened by f xf xsinn xn for all n and

 n

x Then f converges to f butDif f f do es not converge to

n n n n

Dif f f

Thus the names of functions f C do not contain suciently much nitely C

accessible information in order to compute names of the derivatives On the other

C

hand the integration op erator is computable

Theorem computability of integration

The integration op erator Int C IR IR IR IR dened by

b

Z

f xdx Intf a b

a

is computable

IR C C C

We omit the pro of As a corollary the op erator Int C IR where Int f

 

R

f xdxis computable

C C 

Determination of zeros

Determination of zeros

Determination of zeros is an imp ortant task in In this Chapter we

study under which circumstances zeros of functions from C can b e determined

eectively

j

For a continuous function f IR IR the set fx IR j f x g of the nonzeros

is op en and for every op en set X there is a continuous function f IR IR such

that X is the set of nonzeros We prove a computable version of this fact see Defs

Theorem characterization of the set of zeros

Let S ff X C IR j f fg IR n X g Then

IR

S is computable

IR op

S is computable

op IR

Pro of

Since IR nfg is computable the statement follows immediately from Theo

op

rem

For any p u v dom dene pIR IR by

  op

X

n

px f x

n

n

where

min v x x u if u xv

n n n n

f x

n

otherwise

Then IR n x p fg

op

An easy estimation shows that the function C IR has a

C C

computable apply function By Theorem we obtain ie there is a

IR

computable function g with p g p for all p dom

IR op

Therefore p g p S for all p dom

op op IR

It can b e shown that there is some computable set X IR such that the Leb esgue

op

measure X is less than and x X for every computable real numb er Sp e

C

Wei Wei Therefore by Theorem there is a computable function

with many zeros eg in the interval but without any computable zero As a

consequence the relation R ff x C IR j f xg cannot b e

C IR

computable since computable functions f map computable elements

to computable elements We prove that R is not even continuous

C C

Theorem impossibility of zero nding

Let R ff x C IR j f xgThenR is not

C C

continuous

Pro of

For any x IR dene the p olygon function Gx by the vertices

x x

u

u u

x

u

Dene C by pG p Then the apply function of

C

is computable Since Theorem holds accordingly for we obtain

C C C

ie there is some computable function g with G p g p for all

C C C

p dom

C

Now assume that there is a continuous function h with p hp if p

C C C

has a zero Let q Then q and y hg q is a zero of g q

C C C

G q G Obviously y First we consider the case y There

C

i

is a sequence q in with q and lim q q Since y hg q is

i i C i i i C i

i

i

a zero of g q G q G wehave y for all i Since hg is

C i C i i C

continuous wehave

y hg q hg lim q lim hg q lim y

C C i C i i

i i i

This is a contradiction The case y is handled accordingly

Notice that even the very small subset R R fGx j x RgIR is not

Determination of zeros

continuous The contradiction has b een derived by using the function G

C C

which is zero on an op en interval If we exclude such situations and if we consider

only functions whichchange their sign on we obtain a p ositive result The

following theorem is an eectiveversion of a generalized intermediate value theorem

from classical analysis If f IR is continuous and changes its sign then f

has a zero

Theorem non extensional solution

Let F ff C j x y f x f y and I f fg for no op en

nd

interval I g Let

R ff x F IR j f xg

nd nd

Then

R is computable

nd C C

R has no continuous choice function

nd C C

Pro of

The following observations can b e proved easily

Let f F and a b with f a f b Then there are rational

nd

numb ers a b QI with a a b bb a b a f a f a

f a f b and f b f b

The sets fu p j p u g and fu p j pu g are re in

C C

dom

C

There is a Typ e machine M which for input p dom computes sequences

C

u u and v v of elements of dom and pro duces the output q

  Q

u u dom according to the following rules First M searches for

 C

u pv Assume u and v have b een words u v such that p

 C  n n   C

u u v v determined Then M searches for words u v with

n n n n n n

v u v u pu pu pu pv

n n n n C n C n C n C n

and pv pv

C n C n

F is the input for M By the ab ove observations M determines Assume p

nd

C

some q u u dom Let f pandx q We prove

 C C C

u lim v x Consider the case f u Then f x Wehave lim

i i 

i i

f u and f v for all i Bycontinuityof f wehave f x

i i

u lim f u andf xf lim v limf v therefore f lim

i i i i

i i i i

u we obtain f x corresp ondingly f xIf f



Supp ose that there is a continuous function Z C IR such

C C

that fZf for all f F For x IR let Gx b e the p olygon func

nd

tion with the vertices x x Then G IR

C is computable hence continuous Therefore the func

C C C C

tion ZG IR IR is continuous ie continuous by Theorem

C C

and has the prop erty that ZGx is a zero of Gx for all x IR Since conti

nuous functions map intervals onto intervals I ZG since

ZG I and I and I ZG since



I and ZG I This is contradiction

 

The following example illustrates Theorem

Example

Consider the problem to determine a zero of the function f IR IR from a

a

given number a where



f x x x a

a

for all x IR Since for a the zeros of f are in the interval we

a

may restrict the domains to and assume f C for all a

a



f















      

Determination of zeros

By the metho d describ ed in of the pro of of Theorem for given a

one determines sequences of rational numb ers a and b with

i i i i

a a b b fa fb b a b a

i i i i a i a i i i i i

i

The sequence a converges with sp eed to a zero x of f If f has zeros

i i a a a

then it may dep end on the given name p fag which zero the leftmost or the

C

rightmost is determined For every algorithm such dep endence on the names must

o ccur since there is no continuous function Z IR with f Z a for

a

all a The pro of is quite similar to that of Theorem

As a corollary of Theorem we obtain a computable version of the intermediate

value theorem

Corollary

Let F ff C j f is increasing and f f g The function

iv

Z C IR with domZ F and Z f the zero of f is

iv

computable

C C

The function Z from this corollary can b e extended to all continuous functions which

ve exactly one zero ha

Theorem

Let F ff C j f has exactly one zerog The function Z

C IR with domZ F and Z f the zero of f is

C C

computable

Pro of

Let b e some standard numb ering of Let M be a Typ e machine



j j

which for input p w c w c dom pro duces a sequence q u v u v

 C  

such that

k

uv and u v if i uv with



k

u v

jw x j for all x n u v

i i

k

otherwise

Then p f p whenever p F

C I M C

Corollary

If f C is computable and x is an isolated zero of f then x is

computable

Pro of

Assume x Then there are rational numb ers rs with rxs

such that x is the only zero of f in r s Dene f C by

f r if yr

f y if r y s

f y

f s if s y

div otherwise

Then f is computable cf Lemma and x is its only zero By Theorem we

have x Z f Since Z is computable and f is computable x Z f

C C C

is computable

C

Although there is no general metho d of determining zeros for continuous functions

it is p ossible to determine for f C and n some x IR even x QI with

n

jf xj provided f has a zero

Theorem approximate zero

The relation

n

R ff n s C QI jjf sj g

computable is

bin Q C

Pro of

j j

There is a Typ e machine M which for inputs p w c w c dom and n

 C

n

uj Assoonasthe searches for some k and u dom with jw

Q k

search has b een successful M gives u as its output

For every continuous increasing function f C IR the inverse function f is

Determination of zeros

continuous Weprove a computational version of this theorem for simplicityonly

for functions f with r ang ef IR generalizations are straightforward

Theorem inverse function

The function Inv C IR C IR with

f if f is increasing and r ang ef IR

Invf

div otherwise

is computable

IR IR

Pro of

We generalize the metho d for determining zeros of continuous increasing functions

Since x y x y is computable on IR by Theorem the function H

R with H f x y f x y is computable C IR IR IR I

IR C C C

Let M be a Typ e machine which for inputs p dom and q dom

IR C

computes a sequence r u u as follows For determining u M searches

 n

u q H p v q and for u v dom such that H p

C IR C Q IR

n

v u As so on as the search has b een successful M cho oses u u Since

n

H is computable the search can in fact b e programmed byaTyp e machine The

search is successful for every n and q dom if p is increasing and has

C IR

the range IR Consider f p domInvand y q Then

IR C

f f p q y ie f y f p q

C M C M

Rby p p Then p q f p q Dene C I

IR C C M

ie the applyfunction of is computable By Theorem we obtain

C C

This means that there is a computable function g with

IR

Inv p p g p

IR IR

for all p with p domInv Therefore Inv is computable

IR IR IR

While for functions from C maximum values can b e computed by Theorem

the determination of maximum p oints is as dicult as the determination of zeros

This follows from the following observation

x is a zero of f ix is a maximum p ointof g where g xjf xj

x is a maximum p ointoff ix is a zero of h where hxf x Maxf

Notice that Max is computable by Theorem and that computabilityofx y

x y and x jxj can b e derived from Theorem

Computation Time and Lo okahead

on

Time and tap e complexity are the most imp ortant computational complexitymea

sures for Turing machine computations They mo del time and storage requirement

of digital computers quite realistically In this section weintro duce the time com

m

plexity for Typ e machines M with f As a further imp ortant

M

concept we dene the input lookahead which measures the amount of information

which is used during a computation We prove that core sets are the natural

classes with uniform time b ound

m

Let M beaTuring machine with f The computation time of M

M

for input x x is dened by

m

T ime x x the numb er of computation steps which M with input

M m

x x needs until it reaches a HALT statement

m

A function t is a time b ound for M i

m

T ime x x tmax lg x for all x x

M m i m

i

Example

Consider the multiplication of natural numb ers in binary notation Using the scho ol

metho d a Turing machine M can b e constructed such that

f u v u v for all u v dom

bin M bin bin bin



T ime u v cn c where n maxlg ulgv and c is a constant

M



Therefore M multiplies binary numb ers in time t for some t O n

m

Rememb er for f

m m

x g x cf xcg O f fg j c

For t

TIMEt ff j M is a Turing machine and some t O t

M

is a time b ound for M g

is the complexity class of functions computable on Turing machines in Time O t

The ab ove denition of Time cannot b e used for machines with innite output

M

since valid computations never reach a HALT statement Weintro duce as a further

Computation Time and Lo okahead on

parameter a number k and measure the time until M has pro duced the output

symbol q k of its innite output q Another imp ortant information is the input

lookahead ie the numb er of input symb ols which M requires for pro ducing the

m

output sequence q q k In the following we consider only the case Y

for some m

Denition time and input lookahead

m m

Let M beaTyp e machine with f For all y

M

and k dene time and input lookahead by

T ime y k the numb er of steps which M with input y

M

needs until the k th output symb ol has b een written

Ila y k the maximal j such that M with input y

M

reads the j th symb ol from some input tap e during

the rst Time y k computation steps

M

Notice that T ime y k may exist for some but not for all k in such a case

M

y domf Since reading an input symb ol requires at least one computation step

M

Ila y k T ime y k The input lo okahead Ilay is a mo dulus of

M M

m

in the p oint y domf continuity of the function f

M M

While for a Turing machine T ime y is a natural numb er for any y domf for

M M

m

aTyp e machine M with f the function T ime y

M M

determines the computation time of M with input y domf as a function of the

M

output precision and Ila y determines the amount of input information

M

used by M with input y domf as a function of the output precision

M

For anyTyp e machine M the prop erties T ime y k t and T ime y k t

M M

are decidable and the prop erties Ila y k t and Ila y k t are re in

M M

y k t A simple comterexample shows that Ila y k t and Ila y k t

M M

are not recursive in general We shall consider b ounds for time and input lo okahead

m m

which are uniform for all y X for some X The sets X such

that Time y has a computable b ound uniform for all y X can b e characterized

M

m m

easily A set X is called core i n X is re

Theorem uniform time on core sets

m

Let M be a Typ e machine with f

M

If X domf is core then y X k Time y k tk for

M M

some computable function t

If t is computable then

m

X fy j k Time y k tk g

M

is core and X domf

M

Pro of

For simplicitywe consider only the case m The general case is proved accor

dinglyWe use the imp ortant fact that the metric space d is compact

Since X is core there is some Typ e machine N with f such

N

that n X domf Consider k Thenforany p there is some n

N

suchthat

T ime pn or Time pk n

N M

Let n b e the rst such n and w the prex of p of length n Since

p p p

fw j p g and is compact there is a nite set A with

p

fw j p Ag Determine from k anumber tk as follows Search

p

for a nite set W of words with fw j w W g and T ime w

N

lg w orT ime w k lg w for all w W By the ab ove considerations

M

such a set W exists Dene tk maxflg w j w W and Time w k

M

lg w gThen Time pk tk for all p X The function t is

M

computable

Let t b e computable There is a Typemachine N which halts for

pk tk for some k Thendomf n X input p iT ime

M N

We shall call a sequence p computable in time t ithereisaTyp e



machine M with f suchthat f p and Time k tk

M M M

for all k

Computational ComplexityofRealFunctions

Computational Complexity of Real

Functions

In this section weintro duce a new representation of the real numb ers for measuring

the time complexity of real functions Weprove b ounds of time and input lo okahead

for addition multiplication and as an application of Newtons metho d inversion

Finally we discuss the computational complexity of compact sets

By the Main Theorem a real function is continuous i it is determined bya

continuous function on names By denition a real function is computable i

C

it is determined by a computable function on names Wewould like to call a

C

real function computable in time t i it is determined by a function on

names computable in time t

C

Unfortunately this denition is unreasonable First we observe that any name p

C

ofanumber x IR can b e padded arbitrarily Assume p u u and px

 C

and let r b e some function Then some q w w with q x

 C

can b e determined easily such that lg w r i for all i cho ose w dom

i i Q

i

w u j Let M with very large numerator and denominator such that j

i i

beaTyp e machine which computes a real function g IR IR on names

C

By padding the outputs of M a machine M can b e constructed which computes g

on names and op erates in time O n Therefore every computable real function

C

can b e computed in time O non names

C

Toavoid this degeneracy dene temp orarily g is computable in time t

i some Typ e machine M computes g on names such that g x is determined

C

k

IR is with error in at most tk steps But not even the identity id IR

computable in time t for any t since on the input tap e arbritrarily

redundant ie padded names are allowed

We solve the problem byintro ducing a new representation IR of the

real numb ers with which do es not allow padding This representation is a

C

generalization of the representation by innite binary fractions in which additionally

the digit may b e used We shall denote the digit by

Denition modied binary representation

Dene IR as follows where denotes the digit

dom fa a a a j n a f g for i n

n   i

j

a ifn anda a f g if n g

n n n

i

a a a a fa j i ng

n   i

Let p a a a a domandpk a a a a for k

n   n  k

Then

k

pk z for some integer z ZZ

and for this number z

k k

pk z z z z

k

pk z z

k

pk z z

k

pk z z

Therefore p determines a sequence I of nested closed intervals I pk

k k k

such that

the middle half I if a and the right I is the left half of I if a

k k k k k

half of I if a

k k

k

length I

k

pfI j k g

k

which can For reducing redundancy wehave excluded the prex the prex

b e replaced by and the prex which can b e replaced by Although the repre

sentation is not injective no representation equivalentto can b e injectiveby

C

Theorem the sets X for compact X IR and esp ecially the sets fxg

x IR are compact ie small Rememb er that a subset X of the Cantor

space is compact i it is closed

Theorem

C

For any compact subset X IR X is compact

For any computable subset X IR X is core see Def

w

Pro of

Translators from to and vice versa can b e programmed easily

C

b e a sequence in X converging to some p Since X is Let p

i i

b ounded there is some k such that each p has the form w q with lg w

i i i i

k We conclude p dom The representation is continuous since is

C

continuous and Bycontinuity p p dom implies p p

C i i

X hence Since p X for all i and since X is closed we obtain p

i

p X Therefore X is closed and compact

We leave the pro of to the reader

Computational ComplexityofRealFunctions

By Theorem weknow that the time of a Typ e machine is uniformly b ounded

by a computable function on any core subset of its domain which is esp ecially

compact By Theorem every computable real function has a uniform

computable complexity b ound on every computable subset of its domain

w

Denition

m

Let f IR IR b e a computable function let X domf and let

s and t b e functions

ATyp e machine M computes f on X in time t with input lo okahead s

i

f p p f p p

m M m

n tn Time p p

m M

Ila p p n sn

M m

for all n whenever p p X

m

As a rst example we consider addition on IR

Lemma addition

There is a Typ e machine op erating in time O k with input lo okahead

k such that

f p q pq

M

for all p q f g

Pro of

Consider p a a and q b b a b f g Dene r a b For

  i i

g such that n cho ose inductively r f g and c f

n n

r a b c r

n n  n  n n

If c with jr j exists then c and r with jr j exist By induction

n n n n n

j

c and r exist for all n If c dene f p q c c if c dene

n n   

f p q c c c Obviously there is a Typ e machine M which pro duces

 

f p q in time O n with input lo okahead n Weprove the correctness of M

By induction one shows easily

X X X

i i i n

a b c r

i i i n

in  in  in

for all n Consequently pq f p q

Theorem addition



For every b ounded subset X IR there are constants c and c suc hthat



addition on X can b e computed wrt bya Typ e machine in time

c n c with input lo okahead n c



Pro of

m m  m m

There is some m suchthatX If p then p wq

for some w f g with lg w m Let M be a Typ e machine which for

input p q withpq X shifts the p oints in p and qm p ositions to

the left runs the machine from lemma and shifts the p oint of the result m

p ositions to the right

We reduce the multiplication of real numb ers wrt to multiplication of binary

integers by a doubling metho d For obtaining go o d time estimations we need regular

time b ounds FS Mue As a to ol we use the following improvement lemma

whichwe do not prove here

Lemma improvement lemma

j

b Then there are Let I ua a vb

m k m

c c f g with I ua a c c A word

m m k m m m k

c c can b e determined from u v a a and b b in time

m m k m m k

O n where n lg v m k

We shall call a function f regular i

j

f is nondecreasing and nf nand

there are numb ers n c with



tn tn ctn for all n n 

Computational ComplexityofRealFunctions

We state without pro ofs see Mue that for regular functions t

n O t

k

t O n for some k

tcn c O t for every c

P

k

ft j k dlog neg O t





Most of the commonly used b ounds for complexity classes like p olynomials n log n

n log n log log n are regular In the following let Mb be any regular

upp er time b ound for binary integer multiplication on Turing machines For example

bySchonhages metho d Sch n log n l og l og n is such a b ound

Lemma multiplication

There is a Typ e machine N op erating in time O Mb with input lo okahead

n such that

f p q p q

N

g for all p q f

Pro of

Consider p a a and q b b a b f g N pro duces the output

  i i

sequence r c c in stages as follows



Stage

Let x a b y b b Dene c fifx y ifx y

      

if x y g

 

Stage n n

n

Let k N multiplies the nite generalized binary fractions a a and

k 

b b and rounds the result to e e Then according to lemma N

k  k

improves the result c c from Stage n with e e to c c

k k

k

We prove the correctness of the machine N Dene x p y q x

m

a a y b b for m

m m m

The denition of c guarantees a a b b c hence xy

 

c

n

Consider n and k Ife e is a rounding of x y then

k k  k 

k

je e x y j

k k  k 

Furthermore

jxy x y j jx x jjy j jx jjy y j

k  k  k  k  k 

k  k  k 

k k 

k

The triangle inequalityyields je e xy j hence xy e e

k k

An induction with application of Lemma shows xy c c For determining



n n

c N uses the symb ols a and b for determining the symbols c for i

i

n

N uses the symb ols a and b with j Therefore N works with input lo oka

j j

head k We estimate the computation time for Stage n Since a a

k 

k 

can b e written as u v with lg ulgv k N can determine

bin bin

the pro duct x y in at most c Mbk c steps The other computations require

k  k 

at most c k c steps Therefore for any m the word c c is determined

  m

by N in at most

X

i i

sm fc Mb c c c j i dlog meg

 

steps Since Mb is regular c Mbj c c j c O Mb and again by

 

yof Mb s O Mb regularit

By reduction to Lemma one proves easily

Theorem multiplication



For every b ounded subset X IR there is a Typ e machine M which

p erforms multiplication on X in time O Mb with input lo okahead n c

for some constant c

The ab ovemultiplication algorithm uses a doubling metho d The time can b e

 dlog ne

b ounded by tn f f f If f is regular then t O f

A general case where a doubling metho d can b e used is Newtons metho d for deter

mining zeros

By Newtons metho d a zero y of a function f is determined as the limit of a

sequence x where x x f x f x If in some neighbourhood of y

n n n n n n

j

f x and f x is b ounded the sequence x converges quadratically if

n n

x is suciently near to y We consider the computation of x x as a simple but

 

imp ortant example For a letf xx a Then a is the zero of f Simple

computations showthatx x ax is the Newton recursion equation in

n n n



this case and that jx aj jaj jx aj quadratic convergence

n n

Computational ComplexityofRealFunctions

Lemma inversion

There is a Typ e machine M op erating in time O Mb with input lo oka

head k for k and k for k and

f pp

M



for all p f g f g

Pro of

Consider p a a ATyp e machine M pro duces the output sequence r



c c in stages according to the following rules



Stage

From a a determine c c suchthat

 

x a a x c c

 

Dene z c c

 

Stage n n

n

k r a a y z r z

n n k  n n n n

n

e e a rounding of y to k digits

k n n

n

z e e

n k

n

Let c c be the improvement of the result from Stage n with e e

k k

n n

by Lemma

Wehave to prove the correctness of the machine and to make time and input lo oka

head estimations By the restriction for p weha ve p Let a p

If x then x The interval I a a has



length A simple numerical calculation shows that its image J wrt x x



Therefore digits c c exist with a J c c has length

 

a a Asa The machine M contains a nite table for determining c c from

 

 k 

result jz aj where k Consider n and assume that

 

k

n

z e e has b een determined such that jz aj If

n k n

n

k k  k 

n n

n

x z az then jx ajjaj Since jr aj

n n n n n

 k  k 

n n

and jz j since a jx y jz By the rule for

n n n

n

k

n

rounding jy z j We obtain jz ajjz y j jy x j jx aj

n n n n n n n n

k

n

k

n

therefore c c a aj By induction jc c

 k

n

We estimate the input lo okahead of the machine Simple numerical estimations show

that c can b e determined from a a c c from a a and c c c from a a Fur

    

thermore c c is determined from a a and for n e for k j k

  j n n

is determined from a a From this we conclude that the input lo okahead of

k 

n

M is k if k and k ifk For counting input lo okaheads observe

that the input and the output b egin with Since Mb is regular Stage n can b e

n

computed in c Mb c steps Summation yields a time b ound in O Mbfor M

Theorem inversion

For every compact subset X IR with X there is a Typemachine M

which computes x x on X in time O Mb and input lo okahead n c

where c dep ends on X

Pro of outline

m m

There is some m suchthat jxj for all x X We consider the case

x wlg Assume px Then the rst digit of pwhich is dierent from

is By at most m applications of the transformations and

z

from p some z ZZ with jz j m and q a a a q with q x

 

and a a a f g can b e determined Some r with r q can b e

 

determined in Mbn time with lo okahead k c by Lemma Finally the binary

pointof r is shifted by z p ositions

As a nal application we dene recursiveness and computational complexity for

n

subsets X IR Asubset A is recursive i the characteristic function cf

A

cf xifx A otherwise is computable The direct generalization

A

to subsets of IR X IR is recursive i its characteristic function cf IR is

X

computable is useless since by Theorem cf and cf are the only characteristic

IR

functions which are computable If we consider as a metric subspace of the real

j

line a subset A A is recursive i the function d IR is

A bin

computable where d xminfjx ajja Ag This characterization has a useful

A

generalization

Denition complexity of compact sets

n

j

R A and A compact dene For any A I

n

d IR IR by d xinffjx ajja Ag

A A

A is recursive i d is computable

A

A is computable in time ti d is computable in time t

A

n

n

Simple subsets of IR such as the cub e the unit ball every ball with compu

table centre and computable radius as well as its sphere and every convex p olygon

with computable vertices are computable We mention without a pro of that for

Computational ComplexityofRealFunctions

j

A IR A and A compact A is recursive i A is computable where is the

representation from Denition

This denition of recursive corresp onds to located in constructive analysis BB

n

The function d IR IR of A may b e called the lo calizer of AIfn any

A

Typ e machine computing d can b e used by a plotter for pro ducing approximate

A



pictures of the gure A IR Let M b e some Typ e machine computing d

A





IR IR for some compact set A Supp ose wehave a screen divided into

n n n

pixels For i j f g the plotter determines the colour of the pixel

n n n n

P i i j j as follows By simulating the machine M

ij

n n

it computes rational numbers a and b such that d i j

A

n n

a b and b a The pixel P is set to black if a to white

ij

otherwise The construction guarantees

j

A P

ij

P is black

ij

j

P A for some i j with ji i j and jj j j

i j

a

p

ij

u

b

The pixel P is set to black if the annulus contains some p oint x A

ij

S

Consequently the nth approximation A fP is blac kg of A covers A ie

n ij

A A but it surrounds A very narrowly since a pixel P is white if neither the

n ij

pixel itself nor any of its immediate neighb ours intersect A In fact the Hausdor

n

distance d A A is not greater then

H n

The kind of computational complexity of real functions intro duced here is sometimes

called bit complexity Manyinteresting results on bit complexity of real functions

have already b een obtained see eg Bre KF Ko Mue Mue Sch

Other Approaches to Eective Ana

lysis

The approach to computability in analysis presented in this pap er TTE connects

abstract analysis with Turing machine computability Computability is dened ex

plicitly on nite and innite sequences of symb ols Computable functions turn out to

b e continuous Computability and continuity are transferred to other sets by means

of notations and representations where sequences serve as names of ob jects Admis

sible representations which formalize the concept of approximating sequences lead

to very natural computabilityonvarious sets used in analysis The basic machine

mo del admits to intro duce realistic computational complexity in analysis

As already mentioned there are several other approaches to study eectivity in ana

lysis some of which are listed in the following

Numerical analysis can b e considered as the oldest discipline with this aim To

daynumerical algorithms are usually programmed in FORTRAN ALGOL

and realized on computers Such realizations can at most approximate the intended

real functions since they op erate on the nite set of oating p ointnumb ers sup

plied by the machines No mathematical theory of computability or computational

complexity is used

The real RAM real random access machine is a mathematical machine mo del

formalizing the intuitive concept of algorithm used in numerical analysis and com

putational geometry BSS PS Since many TTEcomputable functions are

not real RAMcomputable and since there are real RAMcomputable functions

which are absolutely not computable bya physical device see Lemma the real

RAM mo del is certainly not adequate for generalizing Churchs computability thesis

from the natural num b ers to the real numb ers Sma Noncontinuous functions

computable by real RAMs can b e ordered by levels of discontinuity and classied

by degrees of discontinuity HW

Interval Analysis controls errors which usually o ccur if oating p ointnumb ers are

used for p erforming real computations Mo o Ab e Although no formal de

nition of computability is considered it is very closely related to a denition of

computable real functions given by Grzegorczyk Grz by means of computable

functions on intervals

A computational mo del extending the real RAM is used in IBC information based

complexity TWW For dening computable op erators functions are inserted

into programs as blackboxes or oracles A typical question in IBC is Howmany

evaluations f x are needed for determining the integral of a function f C for

i

some given class C with precision

PourEl and Richards PR generalize a further characterization of the compu

table real functions a real function is computable i it has a computable uniform

mo dulus of continuity and transforms computable sequences of real numb ers to com

putable sequences of real numb ers given by Grzegorczyk Grz to functions on

Other Approaches to Eective Analysis

Banach spaces They study esp ecially solution op erators of dierential equations

from physics

Logical approaches are another way to formalize eectivity in analysis The metho ds

for proving theorems are restricted to constructive ones esp ecially no indirect

pro ofs are allowed see Bee BR Tro for detailed discussions and further

references A very far advanced theory is Bishops remarkable constructive analysis

Bis BB Most of his concepts can b e transferred to TTE if sets are in

terpreted by adequate naming systems and routines by computable or continuous

functions It should however b e mentioned that such logical approaches do not admit

to dene computational complexity

Computational complexity in analysis has b een investigated in dierentways While

in the real RAM mo del and in the IBC approach one evaluation of a real function is

considered as a single step the bit complexity mo dels count the number of Turing

k

machine op erations for approximating a result with a given error Bre KF

Mue Mue Sch Ko Wei TTE emb eds these denitions into a

general frame

Computable analysis based on Grzegorczyks denition via op erators is sometimes

called the Polish approach There is another denition intro duced by Ceitin Cei

Kus Ab e called Russian approach The Russian approach considers only

computable real numb ers Computabilityisintro duced by an eective notation

We explain this more precisely in terms of TTE For any w let b e the

w

function f computed bytheTyp e machine with program w see

App endix B For any representation M of a set M wederive a notation

M of the set M of the computable elements of M Def by

w

w

Wemaysay that w is the element x M computed by the program w relative

to the representation

IR b e the representation from Def Then IR is the set of Let

computable real numb ers In the Russian approach a function f IR IR is

called computable i it is computable Corresp ondingly computabilityis

intro duced on other sets like the re subsets of IR and the computable elements of

C The underlying representations are not dened explicitly but used implicitly

We discuss the relation b etween the Polish und the Russian approach Let

M b e a representation and M the derived notation With

f M M by each function f M M we can asso ciate a function

graphf graphf

The Russian and the Polish approachwould b e essentially equivalent if

f is computable f is computable

The implication can b e proved easily The implication doesnothold

in general but for some imp ortant sp ecial cases

Theorem Ceitin

Let f IR IR b e a function suchthat

X is dense in domf for some re set X dom

Then

f is computable f is computable

Rememb er that computability implies continuity The condition cannot

b e omitted but mightbeweakened The theorem can b e generalized to computable

metric spaces with Cauchy representation Cei KLS Mos Wei

The other case is the MyhillShepherdson theorem MS Weformulate it in

the framework of TTE Let PF fh j h g b e the set of all partial

numb er functions Dene a representation PF of PF by ph i

i j

p enumerates the graph of h more precisely is a subword of p



hi j Notice that PF is the set P of the partial recursive functions and



the standard numb ering of P is equivalentto

Theorem Myhil lShepherdson

 

For any total function f P P

f is computable f is computable

The theorem can b e generalized to computable CPOs cf Wei

Seemingly no other cases in which holds are known Therefore the relation

between the Polish and the Russian approach to computable analysis is not yet

fully understo o d It is wellknown from computability theory that for notations

like the smnfunction is easily computable at most in p olynomial time As

a consequence for each computable function f M M there is

some easily computable function g with f w g w for all

w domf Therefore the Russian approach has no complexity theory

The references given in this pap er esp ecially in Chapter are by no means com

plete Many other authors have contributed considerably to the developmentof

eective analysis I ap ologize to all those whom I did not mention

App endix

App endix A Type machines and their semantics

A Type machine M is dened by

i an inputoutput alphabet and a tape alphabet with andB n

ii a sequence Y Y Y with fY Y gf g sp ecifying the function

k   k

typ e f Y Y Y

M k 

iii nitely many Turing tapeseach with a readwrite head indexed by n

k n

iv a nite owchart F with the prop erties given b elow

Only the following statements are admitted in a owchart F of a Typ e machine

where i n and a

i Rmove the head on Tap e i one p osition to the right

i L move the head on Tap e i one p osition to the left

i a write a on the square scanned by the head on Tap e i

i a binary branching is a the symb ol on the square scanned by the head on

Tap e i

HALT

AdditionallyforTap es i f kg the input tap es only statements i a and

i Rread only oneway input and for Tap e the output tap e only statement

sequences aRwith a write only oneway output are admitted

The semantics of a Typ e machine is dened via computation sequences of con

gurations As for ordinary Turing machines a congur ation of the Typ e machine

M is determined by the lab el of the statement in the owchart F to b e executed

next and the inscription and head p osition for eachTap e i i n A con

guration K is the successor of a conguration K K K i K is obtained

from K by executing the statement at the lab el of K and going to the next lab el

Let the output inscription of a conguration out K b e the longest word w

immediately to the left of the head on Tap e K is a nal conguration i its lab el

has the statementHALT A computation is a nite or innite sequence K K



of congurations with K K i

i i

Y Y Y computed bytheTyp e Now we dene the function f

k  M

machine M

Consider y y Y Y

k k

The initial conguration K y y is determined as follows

k

The lab el is the initial lab el of the owchart

Tap e m m k has the inscription y the remaining squares have the

m

inscription B and the head is p ositioned on the rst square to the left of the

inscription y

m

All the squares on the remaining tap es have the inscription B

App endix

Case Y



For w we dene

f y y w i there is a nite computation K K K such

M k  t

that K K y y K is a nal conguration and w out K

 k t

Case Y



For p we dene

f y y p i there is an innite computation K K suchthat

M k 

K K y y out K is a prex of p for all i and the sequence

 k i

length out K is unb ounded

i i

App endix B Eective naming systems of sets of functions

First weintro duce pairing functions which are a useful to ol also in Typ e compu

tability

Denition B

For k and x a a a a dene

k k

x a a a

 k

For x y and p q dene

xy x y

xp pxx p

pq pq pq

For k and z z dene

k

z z z z z

k k k

The ab ove tuple functions are injective and computable and the pro jections of their

inverses are computable As a generalization of the eectiveGodel numb ering

 ab ab

P Rog Wei weintro duce notations P a b fg

App endix

Denition B

Let FD b e some standard notation of all owcharts of

FD

Typ e machines with one input tap e

ab

For a b fg let P b e the set of all computable functions f

a b

ab ab ab

For a b fg dene the notation P by xisthe

a b

function f computed by the owchart x

FD

ab

The representations have a computable universal function and satisfy the smn

theorem This can b e expressed as follows

Theorem B

ab

Consider P Then

ab

utm

a b

where utm holds i there is a computable function u

a

such that ux y xy for all x dom and y

ab

Theorem expresses the kind of eectivity of the notations For continuous

functions eective representations can b e intro duced A subset of a top ological space

is called a G set i it is a countable intersection of op en sets

Denition B

F ff j f g

g F ff j f

F ff j f fis continuous and domf isopeng

F ff j f fis continuous and domf isG g

b b

F b fg is the set of all continuous functions f The sets F

and F represent all continuous functions by the following lemma

App endix

Lemma B

Every continuous function f has an extension in F Every

continuous function f has an extension in F

ab

We dene representations of the function sets F

Denition B

ab ab

For a b fg dene F by

b

x py if q xp with x and p

ab

q y

div otherwise

ab ab

The functions F are in fact surjective and satisfy the follo wing

eectivity theorem

Theorem B

ab

Consider F Then

ab

utm

a b

where utm holds i there is a computable function u

a

suchthat ux y xy for all x dom and y

More details and pro ofs can b e found in Wei

App endix C Notations of and QI 

Denition C the notations of and of QI

bin Q

w is dened by dom fg The notation of

bin bin

k 

and a a a a f g

bin k  k 

QI of the rational numb ers is dened by The notation

Q

u u

Q bin

App endix

for all u dom

bin

u u

Q bin

for all u dom nfg

bin

uv u v

Q bin bin

uv u v

Q bin bin

j

for all u v dom with u vf g suchthat u and

bin bin

v have no common divisor u is undened for all other u

bin Q

We shall write u instead of u for all u don

Q Q

A reasonable notation of the set of the natural numb ers should at least havean

re domain and the test n and upwards and downwards counting should b e

computable on names The class of these notations ordered under reducibilityhasa

maximum the notation

bin

Lemma C eectivity of

bin

For all notations of suchthat dom isrewehave

fw j w g and fu v j u v g are re

bin

Roughly sp eaking is the except for equivalence unique p o orest notation of

bin

with re domain for which the zerotest and counting are computable The pro of is

not dicult we omit it Also the notation can b e characterized by an eectivity

Q

requirement and maximality

Lemma C eectivity of

Q

For all notations QI of the rational numb ers QI wehave

fu v w x j u v w xg

Q bin bin bin

is re in dom

The pro of is very easy

App endix

App endix D Admissible representations

Weintro duce a class of very natural representations called admissibleLetM be a

M

set and let b e a set of subsets of M Wesaythat identies the p oints of

M iM and fQ j x Qg fQ j y Qg x y for all x y M

That means each x M can b e identied by those prop erties Q whichhold

for x

Denition D

M

Let M b e a set and let b e a notation of a set which

identies the p oints of M The standardrepresentation M of

M derived from is dened by

px i fw j x w g Enwp

for all p and x M where

Enwpfa a j a a is a subword of pg

k k

Thus Enwpisthesetofallwords w enumerated by p and p is a

name of xi p enumerates the set of all words w with x w Roughly sp eaking

a name of x is a complete list in arbitrary order p ossibly with rep etitions of those

prop erties Q which hold for xWe illustrate the denition by examples

Example

M IR x w w w x

Q

uxv M IR x w w uv with

M A w w A

bin

M the set of op en subsets of IR

IR

j

O w w uc v with u v O

M p w w is a prex of p

M

Every set which identies the p oints of M is a subbase of a T top ology



on M Engelking Eng and any subbase of a T top ology on M identies



M

is dened from by points on M The top ology

f j g

App endix

where

fQ Q j n Q Q g

n n

is a base of the top ology In Example is the usual top ology of the real

IR

line in Example is the Cantor top ology on The representation and

the top ology generated by as a subbase are very closely related

Theorem D

Let b e the top ology on M generated by the subbase r ang e from

Denition D Then

X X is op en in dom for all X M

is continuous for all functions M

t

By is the nal topology of by is the greatest or p o orest except

for equivalence continuous representation of the T space M In



corresp onds to the smntheorem and to the utmtheorem from ordinary re

cursion theory Theorems and are sp ecial cases of Theorem D

We call representations which are tequivalent to some standard representation ad

missible wrt or admissible

Denition D

Let M b e a top ological T space with denumerable subbase A repre



sentation M of M is called admissiblei

is continuous

t

for all functions M

By Theorem D every T space M withdenumerable base has a admissible



representation which is unique except for tequivalence In Example we obtain

hence is admissible In Example weobtain id hence

C C IR 

id is admissible Let b e the top ology induced by the metric on a separable

 C

metric space M d Then M isa T space with denumerable subbase whichhas



a admissible representation The Cauchy representation for examples see Def

and Def is admissible More details can b e found in Wei Wei

For spaces with admissible representations a function is top ologically continuous

i it is continuous wrt the representations see Def This is stated in Theorem

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